27,385 research outputs found
Random perfect lattices and the sphere packing problem
Motivated by the search for best lattice sphere packings in Euclidean spaces
of large dimensions we study randomly generated perfect lattices in moderately
large dimensions (up to d=19 included). Perfect lattices are relevant in the
solution of the problem of lattice sphere packing, because the best lattice
packing is a perfect lattice and because they can be generated easily by an
algorithm. Their number however grows super-exponentially with the dimension so
to get an idea of their properties we propose to study a randomized version of
the algorithm and to define a random ensemble with an effective temperature in
a way reminiscent of a Monte-Carlo simulation. We therefore study the
distribution of packing fractions and kissing numbers of these ensembles and
show how as the temperature is decreased the best know packers are easily
recovered. We find that, even at infinite temperature, the typical perfect
lattices are considerably denser than known families (like A_d and D_d) and we
propose two hypotheses between which we cannot distinguish in this paper: one
in which they improve Minkowsky's bound phi\sim 2^{-(0.84+-0.06) d}, and a
competitor, in which their packing fraction decreases super-exponentially,
namely phi\sim d^{-a d} but with a very small coefficient a=0.06+-0.04. We also
find properties of the random walk which are suggestive of a glassy system
already for moderately small dimensions. We also analyze local structure of
network of perfect lattices conjecturing that this is a scale-free network in
all dimensions with constant scaling exponent 2.6+-0.1.Comment: 19 pages, 22 figure
Invariance of Structure in an Aging Colloidal Glass
We study concentrated colloidal suspensions, a model system which has a glass
transition. The non-equilibrium nature of the glassy state is most clearly
highlighted by aging -- the dependence of the system's properties on the time
elapsed since vitrification. Fast laser scanning confocal microscopy allows us
to image a colloidal glass and track the particles in three dimensions. We
analyze the static structure in terms of tetrahedral packing. We find that
while the aging of the suspension clearly affects its dynamics, none of the
geometrical quantities associated with tetrahedra change with age.Comment: Submitted to the proceedings of "The 3rd International Workshop on
Complex Systems" in Sendai, Japa
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Packing items from a triangular distribution
We consider the problem of packing n items which are drawn according to a probability distribution whose density function is triangular in shape. For triangles which represent density functions whose expectation is 1/p for p = 3, 4, 5, ..., we give a packing strategy for which the ratio of the number of bins used in the packing to the expected total size of the items asymptotically approaches 1
Combining tomographic imaging and DEM simulations to investigate the structure of experimental sphere packings
We combine advanced image reconstruction techniques from computed X-ray micro
tomography (XCT) with state-of-the-art discrete element method simulations
(DEM) to study granular materials. This "virtual-laboratory" platform allows us
to access quantities, such as frictional forces, which would be otherwise
experimentally immeasurable.Comment: 20 pages, 17 figure
Optimal packing of polydisperse hard-sphere fluids II
We consider the consequences of keeping the total surface fixed for a
polydisperse system of hard spheres. In contrast with a similar model (J.
Zhang {\it et al.}, J. Chem. Phys. {\bf 110}, 5318 (1999)), the Percus-Yevick
and Mansoori equations of state work very well and do not show a breakdown. For
high pressures Monte Carlo simulation we show three mechanically stable
polydisperse crystals with either a unimodal or bimodal particle-size
distributions.Comment: 17 pages, 8 figures, revtex (accepted by J. Chem. Phys.
Characterization of maximally random jammed sphere packings: Voronoi correlation functions
We characterize the structure of maximally random jammed (MRJ) sphere
packings by computing the Minkowski functionals (volume, surface area, and
integrated mean curvature) of their associated Voronoi cells. The probability
distribution functions of these functionals of Voronoi cells in MRJ sphere
packings are qualitatively similar to those of an equilibrium hard-sphere
liquid and partly even to the uncorrelated Poisson point process, implying that
such local statistics are relatively structurally insensitive. This is not
surprising because the Minkowski functionals of a single Voronoi cell
incorporate only local information and are insensitive to global structural
information. To improve upon this, we introduce descriptors that incorporate
nonlocal information via the correlation functions of the Minkowski functionals
of two cells at a given distance as well as certain cell-cell probability
density functions. We evaluate these higher-order functions for our MRJ
packings as well as equilibrium hard spheres and the Poisson point process. We
find strong anticorrelations in the Voronoi volumes for the hyperuniform MRJ
packings, consistent with previous findings for other pair correlations [A.
Donev et al., Phys. Rev. Lett. 95, 090604 (2005)], indicating that large-scale
volume fluctuations are suppressed by accompanying large Voronoi cells with
small cells, and vice versa. In contrast to the aforementioned local Voronoi
statistics, the correlation functions of the Voronoi cells qualitatively
distinguish the structure of MRJ sphere packings (prototypical glasses) from
that of the correlated equilibrium hard-sphere liquids. Moreover, while we did
not find any perfect icosahedra (the locally densest possible structure in
which a central sphere contacts 12 neighbors) in the MRJ packings, a
preliminary Voronoi topology analysis indicates the presence of strongly
distorted icosahedra.Comment: 13 pages, 10 figure
Cell shape analysis of random tessellations based on Minkowski tensors
To which degree are shape indices of individual cells of a tessellation
characteristic for the stochastic process that generates them? Within the
context of stochastic geometry and the physics of disordered materials, this
corresponds to the question of relationships between different stochastic
models. In the context of image analysis of synthetic and biological materials,
this question is central to the problem of inferring information about
formation processes from spatial measurements of resulting random structures.
We address this question by a theory-based simulation study of shape indices
derived from Minkowski tensors for a variety of tessellation models. We focus
on the relationship between two indices: an isoperimetric ratio of the
empirical averages of cell volume and area and the cell elongation quantified
by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for
these quantities, as well as for distributions thereof and for correlations of
cell shape and volume, are presented for Voronoi mosaics of the Poisson point
process, determinantal and permanental point processes, and Gibbs hard-core and
random sequential absorption processes as well as for Laguerre tessellations of
polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data
are complemented by mechanically stable crystalline sphere and disordered
ellipsoid packings and area-minimising foam models. We find that shape indices
of individual cells are not sufficient to unambiguously identify the generating
process even amongst this limited set of processes. However, we identify
significant differences of the shape indices between many of these tessellation
models. Given a realization of a tessellation, these shape indices can narrow
the choice of possible generating processes, providing a powerful tool which
can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their
Applications in Stochastic Geometry and Imaging" in Lecture Notes in
Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense
kGamma distributions in granular packs
It has been recently pointed out that local volume fluctuations in granular
packings follow remarkably well a shifted and rescaled Gamma distribution named
the kGamma distribution [T. Aste, T. Di Matteo, Phys. Rev. E 77 (2008) 021309].
In this paper we confirm, extend and discuss this finding by supporting it with
additional experimental and simulation data.Comment: 10 pages, 5 figure
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