Mean-field approach to Random Apollonian Packing

Unraveling the complexities of random packing in three dimensions has long puzzled physicists. While both experiments and simulations consistently show a maximum density of 64 percent for tightly packed random spheres, we still lack an unambiguous and universally accepted definition of random packing. This paper introduces an innovative standpoint, depicting random packing as spheres closest to a quenched Poisson field of random points. We furnish an efficacious algorithm to ...

It is shown that the numerical data in cond-mat/0608362 are in very good agreement with the predictions of cond-mat/0601573.

A formalism is presented for analytically obtaining the probability density function, (P_{n}(s)), for the random distance (s) between two random points in an (n)-dimensional spherical object of radius (R). Our formalism allows (P_{n}(s)) to be calculated for a spherical (n)-ball having an arbitrary volume density, and reproduces the well-known results for the case of uniform density. The results find applications in stochastic geometry, computational science, molecular biolog...

In this short commentary we provide our comment on the article "Explicit Analytical Solution for Random Close Packing in $d=2$ and $d=3$" and its subsequent Erratum that are recently published in Physical Review Letters. In that Letter, the author presented an explicit analytical derivation of the volume fractions $\phi_{\rm RCP}$ for random close packings (RCP) in both $d=2$ and $d=3$. Here we first briefly show the key parts of the derivation in Ref.~\cite{Za22}, and then p...

The properties of the number of iterations in random sequential adsorption protocol needed to generate finite saturated random packing of spherically symmetric shapes were studied. Numerical results obtained for one, two, and three dimensional packings were supported by analytical calculations valid for any dimension $d$. It has been shown that the number of iterations needed to generate finite saturated packing is subject to Pareto distribution with exponent $-1-1/d$ and the...

Packings of identical objects have fascinated both scientists and laymen alike for centuries, in particular the sphere packings and the packings of identical regular tetrahedra. Mathematicians have tried for centuries to determine the densest packings; Crystallographers and chemists have been fascinated by the lattice packings for centuries as well. On the other hand, physicists, geologists, material scientists and engineers have been challenged by the mysterious random packi...

In this article, we consider a configuration of weighted random balls in $\mathbb{R}^d$ generated according to a Poisson point process. The model investigated exhibits inhomogeneity, as well as dependence between the centers and the radii and heavy tails phenomena. We investigate the asymptotic behavior of the total mass of the configuration of the balls at a macroscopic level. Three different regimes appear depending on the intensity parameters and the zooming factor. Among ...

Analytic approximations for the radial distribution function, the structure factor, and the equation of state of hard-core fluids in fractal dimension $d$ ($1 \leq d \leq 3$) are developed as heuristic interpolations from the knowledge of the exact and Percus-Yevick results for the hard-rod and hard-sphere fluids, respectively. In order to assess their value, such approximate results are compared with those of recent Monte Carlo simulations and numerical solutions of the Perc...

Sphere packings are essential to the development of physical models for powders, composite materials, and the atomic structure of the liquid state. There is a strong scientific need to be able to assess the fit of packing models to data, but this is complicated by the lack of formal probabilistic models for packings. Without formal models, simulation algorithms and collections of physical objects must be used as models. Identification of common aspects of different realizatio...

We have studied the contact network properties of two and three dimensional polydisperse, frictionless sphere packings at the random closed packing density through simulations. We observe universal correlations between particle size and contact number that are independent of the polydispersity of the packing. This allows us to formulate a mean field version of the granocentric model to predict the contact number distribution P(z). We find the predictions to be in good agreeme...