139 research outputs found
Sphere Packing with Limited Overlap
The classical sphere packing problem asks for the best (infinite) arrangement
of non-overlapping unit balls which cover as much space as possible. We define
a generalized version of the problem, where we allow each ball a limited amount
of overlap with other balls. We study two natural choices of overlap measures
and obtain the optimal lattice packings in a parameterized family of lattices
which contains the FCC, BCC, and integer lattice.Comment: 12 pages, 3 figures, submitted to SOCG 201
IST Austria Thesis
We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications
Soft Spheres Make More Mesophases
We use both mean-field methods and numerical simulation to study the phase
diagram of classical particles interacting with a hard-core and repulsive, soft
shoulder. Despite the purely repulsive interaction, this system displays a
remarkable array of aggregate phases arising from the competition between the
hard-core and shoulder length scales. In the limit of large shoulder width to
core size, we argue that this phase diagram has a number of universal features,
and classify the set of repulsive shoulders that lead to aggregation at high
density. Surprisingly, the phase sequence and aggregate size adjusts so as to
keep almost constant inter-aggregate separation.Comment: 4 pages, 2 included figure
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
Optimal and non-optimal lattices for non-completely monotone interaction potentials
We investigate the minimization of the energy per point among
-dimensional Bravais lattices, depending on the choice of pairwise potential
equal to a radially symmetric function . We formulate criteria for
minimality and non-minimality of some lattices for at fixed scale based
on the sign of the inverse Laplace transform of when is a superposition
of exponentials, beyond the class of completely monotone functions. We also
construct a family of non-completely monotone functions having the triangular
lattice as the unique minimizer of at any scale. For Lennard-Jones type
potentials, we reduce the minimization problem among all Bravais lattices to a
minimization over the smaller space of unit-density lattices and we establish a
link to the maximum kissing problem. New numerical evidence for the optimality
of particular lattices for all the exponents are also given. We finally design
one-well potentials such that the square lattice has lower energy
than the triangular one. Many open questions are also presented.Comment: 37 pages. 9 figures. To appear in Analysis and Mathematical Physic
Optimal and non-optimal lattices for non-completely monotone interaction potentials
37 pages. 9 figures. To appear in Analysis and Mathematical Physics.We investigate the minimization of the energy per point among -dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function . We formulate criteria for minimality and non-minimality of some lattices for at fixed scale based on the sign of the inverse Laplace transform of when is a superposition of exponentials, beyond the class of completely monotone functions. We also construct a family of non-completely monotone functions having the triangular lattice as the unique minimizer of at any scale. For Lennard-Jones type potentials, we reduce the minimization problem among all Bravais lattices to a minimization over the smaller space of unit-density lattices and we establish a link to the maximum kissing problem. New numerical evidence for the optimality of particular lattices for all the exponents are also given. We finally design one-well potentials such that the square lattice has lower energy than the triangular one. Many open questions are also presented
Dimension reduction techniques for the minimization of theta functions on lattices
We consider the minimization of theta functions
amongst
lattices , by reducing the dimension of the
problem, following as a motivation the case , where minimizers are
supposed to be either the BCC or the FCC lattices. A first way to reduce
dimension is by considering layered lattices, and minimize either among
competitors presenting different sequences of repetitions of the layers, or
among competitors presenting different shifts of the layers with respect to
each other. The second case presents the problem of minimizing theta functions
also on translated lattices, namely minimizing . Another way to reduce dimension is by considering
lattices with a product structure or by successively minimizing over concentric
layers. The first direction leads to the question of minimization amongst
orthorhombic lattices, whereas the second is relevant for asymptotics
questions, which we study in detail in two dimensions.Comment: 45 pages. 7 figure
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