863 research outputs found
Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results
We achieve a detailed understanding of the -sided planar Poisson-Voronoi
cell in the limit of large . Let be the probability for a cell to
have sides. We construct the asymptotic expansion of up to
terms that vanish as . We obtain the statistics of the lengths of
the perimeter segments and of the angles between adjoining segments: to leading
order as , and after appropriate scaling, these become independent
random variables whose laws we determine; and to next order in they have
nontrivial long range correlations whose expressions we provide. The -sided
cell tends towards a circle of radius (n/4\pi\lambda)^{\half}, where
is the cell density; hence Lewis' law for the average area of
the -sided cell behaves as with . For
the cell perimeter, expressed as a function of the polar
angle , satisfies , where is known Gaussian
noise; we deduce from it the probability law for the perimeter's long
wavelength deviations from circularity. Many other quantities related to the
asymptotic cell shape become accessible to calculation.Comment: 54 pages, 3 figure
From one cell to the whole froth: a dynamical map
We investigate two and three-dimensional shell-structured-inflatable froths,
which can be constructed by a recursion procedure adding successive layers of
cells around a germ cell. We prove that any froth can be reduced into a system
of concentric shells. There is only a restricted set of local configurations
for which the recursive inflation transformation is not applicable. These
configurations are inclusions between successive layers and can be treated as
vertices and edges decorations of a shell-structure-inflatable skeleton. The
recursion procedure is described by a logistic map, which provides a natural
classification into Euclidean, hyperbolic and elliptic froths. Froths tiling
manifolds with different curvature can be classified simply by distinguishing
between those with a bounded or unbounded number of elements per shell, without
any a-priori knowledge on their curvature. A new result, associated with
maximal orientational entropy, is obtained on topological properties of natural
cellular systems. The topological characteristics of all experimentally known
tetrahedrally close-packed structures are retrieved.Comment: 20 Pages Tex, 11 Postscript figures, 1 Postscript tabl
On Random Bubble Lattices
We study random bubble lattices which can be produced by processes such as
first order phase transitions, and derive characteristics that are important
for understanding the percolation of distinct varieties of bubbles. The results
are relevant to the formation of topological defects as they show that infinite
domain walls and strings will be produced during appropriate first order
transitions, and that the most suitable regular lattice to study defect
formation in three dimensions is a face centered cubic lattice. Another
application of our work is to the distribution of voids in the large-scale
structure of the universe. We argue that the present universe is more akin to a
system undergoing a first-order phase transition than to one that is
crystallizing, as is implicit in the Voronoi foam description. Based on the
picture of a bubbly universe, we predict a mean coordination number for the
voids of 13.4. The mean coordination number may also be used as a tool to
distinguish between different scenarios for structure formation.Comment: several modifications including new abstract, comparison with froth
models, asymptotics of coordination number distribution, further discussion
of biased defects, and relevance to large-scale structur
Description of Fischer Clusters Formation in Supercooled Liquids Within Framework of Continual Theory of Defects
Liquid is represented as complicated system of disclinations according to
defect description of liquids and glasses. The expressions for the linear
disclination field of an arbitrary form and energy of inter-disclination
interaction are derived in the framework of gauge theory of defects. It allows
us to describe liquid as a disordered system of topological moments and reduce
this model to the Edwards--Anderson model with large-range interaction. Within
the framework of this approach vitrifying is represented as a "hierarchical"
phase transition. The suggested model allows us to explain the process of the
Fischer clusters formation and the slow dynamics in supercooled liquids close
to the liquid--glass transition point
Space-time defects and teleparallelism
We consider the class of space-time defects investigated by Puntigam and
Soleng. These defects describe space-time dislocations and disclinations
(cosmic strings), and are in close correspondence to the actual defects that
arise in crystals and metals. It is known that in such materials dislocations
and disclinations require a small and large amount of energy, respectively, to
be created. The present analysis is carried out in the context of the
teleparallel equivalent of general relativity (TEGR). We evaluate the
gravitational energy of these space-time defects in the framework of the TEGR
and find that there is an analogy between defects in space-time and in
continuum material systems: the total gravitational energy of space-time
dislocations and disclinations (considered as idealized defects) is zero and
infinit, respectively.Comment: 22 pages, no figures, to appear in the Class. Quantum Gravit
A smooth introduction to the wavefront set
The wavefront set provides a precise description of the singularities of a
distribution. Because of its ability to control the product of distributions,
the wavefront set was a key element of recent progress in renormalized quantum
field theory in curved spacetime, quantum gravity, the discussion of time
machines or quantum energy inequalitites. However, the wavefront set is a
somewhat subtle concept whose standard definition is not easy to grasp. This
paper is a step by step introduction to the wavefront set, with examples and
motivation. Many different definitions and new interpretations of the wavefront
set are presented. Some of them involve a Radon transform.Comment: 29 pages, 7 figure
Volterra Distortions, Spinning Strings, and Cosmic Defects
Cosmic strings, as topological spacetime defects, show striking resemblance
to defects in solid continua: distortions, which can be classified into
disclinations and dislocations, are line-like defects characterized by a delta
function-valued curvature and torsion distribution giving rise to rotational
and translational holonomy. We exploit this analogy and investigate how
distortions can be adapted in a systematic manner from solid state systems to
Einstein-Cartan gravity. As distortions are efficiently described within the
framework of a SO(3) {\rlap{\supset}\times}} T(3) gauge theory of solid
continua with line defects, we are led in a straightforward way to a Poincar\'e
gauge approach to gravity which is a natural framework for introducing the
notion of distorted spacetimes. Constructing all ten possible distorted
spacetimes, we recover, inter alia, the well-known exterior spacetime of a
spin-polarized cosmic string as a special case of such a geometry. In a second
step, we search for matter distributions which, in Einstein-Cartan gravity, act
as sources of distorted spacetimes. The resulting solutions, appropriately
matched to the distorted vacua, are cylindrically symmetric and are interpreted
as spin-polarized cosmic strings and cosmic dislocations.Comment: 24 pages, LaTeX, 9 eps figures; remarks on energy conditions added,
discussion extended, version to be published in Class. Quantum Gra
Disclinations, dislocations and continuous defects: a reappraisal
Disclinations, first observed in mesomorphic phases, are relevant to a number
of ill-ordered condensed matter media, with continuous symmetries or frustrated
order. They also appear in polycrystals at the edges of grain boundaries. They
are of limited interest in solid single crystals, where, owing to their large
elastic stresses, they mostly appear in close pairs of opposite signs. The
relaxation mechanisms associated with a disclination in its creation, motion,
change of shape, involve an interplay with continuous or quantized dislocations
and/or continuous disclinations. These are attached to the disclinations or are
akin to Nye's dislocation densities, well suited here. The notion of 'extended
Volterra process' takes these relaxation processes into account and covers
different situations where this interplay takes place. These concepts are
illustrated by applications in amorphous solids, mesomorphic phases and
frustrated media in their curved habit space. The powerful topological theory
of line defects only considers defects stable against relaxation processes
compatible with the structure considered. It can be seen as a simplified case
of the approach considered here, well suited for media of high plasticity
or/and complex structures. Topological stability cannot guarantee energetic
stability and sometimes cannot distinguish finer details of structure of
defects.Comment: 72 pages, 36 figure
Soap Froths and Crystal Structures
We propose a physical mechanism to explain the crystal symmetries found in
macromolecular and supramolecular micellar materials. We argue that the packing
entropy of the hard micellar cores is frustrated by the entropic interaction of
their brush-like coronas. The latter interaction is treated as a surface effect
between neighboring Voronoi cells. The observed crystal structures correspond
to the Kelvin and Weaire-Phelan minimal foams. We show that these structures
are stable for reasonable areal entropy densities.Comment: 4 pages, RevTeX, 2 included eps figure
Computer investigation of the energy landscape of amorphous silica
The multidimensional topography of the collective potential energy function
of a so-called strong glass former (silica) is analyzed by means of classical
molecular dynamics calculations. Features qualitatively similar to those of
fragile glasses are recovered at high temperatures : in particular an intrinsic
characteristic temperature K is evidenced above which the
system starts to investigate non-harmonic potential energy basins. It is shown
that the anharmonicities are essentially characterized by a roughness appearing
in the potential energy valleys explored by the system for temperatures above
.Comment: 5 pages; accepted for publication in PR
- âŠ