Schreck and C. O'Hern personal communication and 0. Thus, our prediction is interpreted as the upper limit in the range of packings observed with numerical algorithms. Under this scenario, which is consistent with analogous three-dimensional 3D spherical results, packings may exist in the region [ th , Edw ], and our theory is a mean-field estimation of Edw.

This region is very small for spheres but the above evidence indicates that non-spherical particles may pack randomly in a broader range of volumes. The present framework estimates the upper bound for such a range. The shapes of dimers, spherocylinders, ellipsoids are then all shown to increase the density of the random packing to first-order.

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In the case of regular crystal packings, recent mathematically rigorous work has shown in fact that for axisymmetric particles any small deformation from the sphere will lead to an increase in the optimal packing fraction of the crystal This appears only in 3D and is related to Ulam's conjecture stating that the sphere is the worst case scenario for ordered packings in 3D ref.

A full mathematical proof of this conjecture is still outstanding, but so far all computer simulations verify the conjecture. In particular, recent advances in simulation techniques allow to generate crystal packings of a large variety of convex and non-convex objects in an efficient manner 48 , The extensive study of de Graaf et al. The verification for regular n -prisms and n -antiprisms can be extended to arbitrary n using this method, providing an exhaustive empirical verification of the conjecture for these regular shapes.

We remark that a random analogue of Ulam's packing conjecture has been proposed and verified for the Platonic solids apart from the cube in simulations The results presented here support the random version of Ulam's conjecture and might help in investigating this conjecture further from a theoretical point of view.

We believe that our decomposition of various shapes into intersections and overlaps of spheres will be a useful starting point for a systematic investigation of this issue. Our approach can be systematically continued beyond the axisymmetric shapes considered here. For instance, in Fig. The challenge would be to implement our algorithm to calculate the resulting Voronoi excluded volumes that appear in our mean-field theory.

For this, one might also consider a fully numerical evaluation using, for example, graphics hardware Mathematically, we can write the local mechanical equilibrium on a generic non-spherical frictionless particle having k contacts defined by their location r j , normal and force , as:. A local degenerate configuration has a matrix such that. We base our evaluation on two assumptions: i Contact directions around a particle in the packing are uncorrelated, and ii Given one set of contact directions, a particle i is found in an orientation such that the redundancy in the mechanical equilibrium conditions is maximal, that is, is a minimum.

Note that depends on , as only the absolute direction of contact points are chosen, and thus rotating particle i affects the direction and normal of these contacts with respect to particle i. This situation is described in Fig. In this case, the rank is reduced by one unit, and the probability of occurrence of such a situation is large at small aspect ratio, as it just requires that there is no contact on the cylindrical part of the inner particle.

We then extract the average effective number of degrees of freedom , which is the average over the contact directions of the minimal value of : , where denotes the average over contact directions. Finally, the normalization is the volume of J. For a packing with a coordination number distribution Q z k , with average z , the effective d f is: , and the average z follows as. In our evaluation, we use a Gaussian distribution for Q z k , with variance 1.

## Advanced Mean Field Methods

The way we look for the orientation on the unit sphere showing the lowest rank is simply by sampling it randomly with a uniform distribution 10 6 samples. We perform a comprehensive test of the different approximations of the theory using computer simulations of spherocylinder packings Supplementary Note 1. From the generated configurations at the jamming point we obtain the CDF P c , z , where z is also an observable of the simulation determined by the jamming condition.

P c , z contains the probability that the boundary of the Voronoi volume in the direction is found at a value larger than c and is determined as follows. We select an orientation relative to the orientation of a chosen reference particle i.

## Covered topics

A large number of particles in the packing contribute a VB along with particle i. We determine all these different VBs denoted by. The boundary of the Voronoi volume in the direction is the minimum c m of all positive VBs:. Determining this minimal VB for all particles i in the packing yields a list of c m values for a given which is always relative to the orientation. The CDF P c , z simply follows by counting the number of values larger than a specified c. Moreover, due to inversion symmetry it is sufficient to select only. As the packing is statistically isotropic for all azimuthal angles, the resulting c m value for these directions can all be included in the same ensemble.

The results are plotted in Fig. We plot the theoretical predictions solid lines for P c , z black , P B c red and P C c , z green with the corresponding CDFs sampled from simulated configurations symbols of spherocylinders.

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We generally observe that the three CDFs agree quite well in the regime of small c values, which provides the dominant contribution to the average Voronoi volume. The same plots are shown on a linear scale in the Supplementary Fig. The error bars denote the root mean square error of the finite-size sampling.

We test the two main approximations considered in the theory: a The derivation of P c , z using a liquid like theory of correlations as done in Song et al. This approximation neglects the correlations between the contacting particles and the bulk. In Fig. In order to determine the P B c from the simulation data we need to take the contact radius between particle i and any particle j into account. The minimal VB, c m , is determined from the contributed VBs of particles in the bulk only, that is, particles with. Likewise, P C c is determined from the simulation data by only considering VBs of contacting particles with.

Following this procedure, we have tested these approximations with the computer generated packings. We find Fig. The small values of c provide the dominant contribution in the self-consistent equation to calculate the average Voronoi volume equation 4 , and therefore to the main quantity of interest, the volume fraction of the packing. This can be seen by rewriting equation 4 as. The main contribution to the integral then comes from c values close to due to the decay of the CDF.

Systematic deviations in our approximations arise in the bulk distribution P B for larger values of c , but, interestingly, the slope of the decay still agrees with our theory. Overall, the comparison highlights the mean-field character of our theory: Correlations are captured well up to about the first coordination shell of particles, after which theory and simulations diverge, especially for the bulk term.

The agreement is acceptable for the nearest neighbour-shell, but is incorrect for the second neighbours. Beyond this shell, bulk particles are affected in a finite range by correlations that we do not address, as we assume a uniform distribution of the density of these particles; this is a typical assumption in a mean-field theory.

The additional unaccounted correlations lead to a slightly higher probability to observe the VB at intermediate c values in the simulation, compared with our theory. However, these deviations from simulations are small. For instance, Fig. This small discrepancy is not relevant, as such a value of the probability is negligibly small in the calculation of the volume fraction in equation 4. These results indicate that, overall, the theory captures the distribution of VBs in the region of small c , which is the relevant region in the calculation of the volume fraction.

The neglected higher-order correlations in the upper coordination shells can only decrease the volume fraction in the calculation leading to smaller packing densities. Following this analysis, we interpret our predicted packing fractions as upper bounds for the empirically found ones, which is indeed observed in Fig. How to cite this article: Baule, A. Mean-field theory of random close packings of axisymmetric particles. Glotzer, S. Anisotropy of building blocks and their assembly into complex structures.

Damasceno, P. Predictive self-assembly of polyhedra into complex structures. Science , — Ni, R. Phase diagram of colloidal hard superballs: from cubes via spheres to octahedra. Soft Matter 8 , — Williams, S. Random packings of spheres and spherocylinders simulated by mechanical contraction. E 67 , Abreu, C. Influence of particle shape on the packing and on the segregation of spherocylinders via Monte Carlo simulations. Powder Technol. Donev, A.

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Improving the density of jammed disordered packings using ellipsoids. Man, W. Experiments on random packings of ellipsoids. Jia, X. Validation of a digital packing algorithm in predicting powder packing densities. Bargiel, M. Geometrical properties of simulated packings of spherocylinders. Wouterse, A. On contact numbers in random rod packings. Granular Matter 11 , — Haji-Akbari, A.

Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra. Nature , — Faure, S. Jaoshvili, A.

Experiments on the random packing of tetrahedral dice. Lu, P. A computational investigation on random packings of sphere-spherocylinder mixtures.

## Advanced Mean Field Methods | The MIT Press

Science China 53 , — Kyrylyuk, A. Isochoric ideality in jammed random packings of non-spherical granular matter. Soft Matter. Jiao, Y. Maximally random jammed packings of platonic solids: hyperuniform long-range correlations and isostaticity. E 84 , Zhao, J. Dense random packings of spherocylinders. Torquato, S. Jammed hard-particle packings: From Kepler to Bernal and beyond. Parisi, G. Mean-field theory of hard sphere glasses and jamming. Song, C. A phase diagram for jammed matter. Aurenhammer, F. Voronoi diagrams - a survey of a fundamental geometric data structure.

ACM Computing Surveys 23 , — Okabe, A. Boissonat, J. Phillips, C. Optimal filling of shapes. Hoff, K. Fast computation of generalized voronoi diagrams using graphics hardware. Makse, H. A Bradford Book. This book covers the theoretical foundations of advanced mean field methods, explores the relation between the different approaches, examines the quality of the approximation obtained, and demonstrates their application to various areas of probabilistic modeling.

A major problem in modern probabilistic modeling is the huge computational complexity involved in typical calculations with multivariate probability distributions when the number of random variables is large. Because exact computations are infeasible in such cases and Monte Carlo sampling techniques may reach their limits, there is a need for methods that allow for efficient approximate computations.

One of the simplest approximations is based on the mean field method, which has a long history in statistical physics. The method is widely used, particularly in the growing field of graphical models. Researchers from disciplines such as statistical physics, computer science, and mathematical statistics are studying ways to improve this and related methods and are exploring novel application areas. While the analysis of MCMC methods reposes on the theory of Markov chains and stochastic matrices , mean-field methods make links to optimization theory and perturbation theory.

Underlying much of the heightened interest in these links between statistical physical and the information sciences is the development in the latter field of a general framework for associating joint probability distributions with graphs, and for exploiting the structure of the graph in the computation of marginal probabilities and expectations. Categories : Publication Publication Facts about " AdvancedMeanFieldMethods ". RDF feed.

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