105,056 research outputs found
Uniform line fillings
Deterministic fabrication of random metamaterials requires filling of a space
with randomly oriented and randomly positioned chords with an on-average
homogenous density and orientation, which is a nontrivial task. We describe a
method to generate fillings with such chords, lines that run from edge to edge
of the space, in any dimension. We prove that the method leads to random but
on-average homogeneous and rotationally invariant fillings of circles, balls
and arbitrary-dimensional hyperballs from which other shapes such as rectangles
and cuboids can be cut. We briefly sketch the historic context of Bertrand's
paradox and Jaynes' solution by the principle of maximum ignorance. We analyse
the statistical properties of the produced fillings, mapping out the density
profile and the line-length distribution and comparing them to analytic
expressions. We study the characteristic dimensions of the space in between the
chords by determining the largest enclosed circles and balls in this pore
space, finding a lognormal distribution of the pore sizes. We apply the
algorithm to the direct-laser-writing fabrication design of optical
multiple-scattering samples as three-dimensional cubes of random but
homogeneously positioned and oriented chords.Comment: 10 pages, 12 figures; v3: restructured paper, more references, more
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Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection
By utilizing diffusion maps embedding and transition matrix analysis we
investigate sparse temperature measurement time-series data from
Rayleigh--B\'enard convection experiments in a cylindrical container of aspect
ratio between its diameter () and height (). We consider
the two cases of a cylinder at rest and rotating around its cylinder axis. We
find that the relative amplitude of the large-scale circulation (LSC) and its
orientation inside the container at different points in time are associated to
prominent geometric features in the embedding space spanned by the two dominant
diffusion-maps eigenvectors. From this two-dimensional embedding we can measure
azimuthal drift and diffusion rates, as well as coherence times of the LSC. In
addition, we can distinguish from the data clearly the single roll state (SRS),
when a single roll extends through the whole cell, from the double roll state
(DRS), when two counter-rotating rolls are on top of each other. Based on this
embedding we also build a transition matrix (a discrete transfer operator),
whose eigenvectors and eigenvalues reveal typical time scales for the stability
of the SRS and DRS as well as for the azimuthal drift velocity of the flow
structures inside the cylinder. Thus, the combination of nonlinear dimension
reduction and dynamical systems tools enables to gain insight into turbulent
flows without relying on model assumptions
Transport in time-dependent dynamical systems: Finite-time coherent sets
We study the transport properties of nonautonomous chaotic dynamical systems
over a finite time duration. We are particularly interested in those regions
that remain coherent and relatively non-dispersive over finite periods of time,
despite the chaotic nature of the system. We develop a novel probabilistic
methodology based upon transfer operators that automatically detects maximally
coherent sets. The approach is very simple to implement, requiring only
singular vector computations of a matrix of transitions induced by the
dynamics. We illustrate our new methodology on an idealized stratospheric flow
and in two and three dimensional analyses of European Centre for Medium Range
Weather Forecasting (ECMWF) reanalysis data
Critical Phenomena in Quasi-Two-Dimensional Vibrated Granular Systems
The critical phenomena associated to the liquid to solid transition of
quasi-two-dimensional vibrated granular systems is studied using molecular
dynamics simulations of the inelastic hard sphere model. The critical
properties are associated to the fourfold bond-orientational order parameter
, which measures the level of square crystallization of the system.
Previous experimental results have shown that the transition of , when
varying the vibration amplitude, can be either discontinuous or continuous, for
two different values of the height of the box. Exploring the amplitude-height
phase space, a transition line is found, which can be either discontinuous or
continuous, merging at a tricritical point and the continuous branch ends in an
upper critical point. In the continuous transition branch, the critical
properties are studied. The exponent associated to the amplitude of the order
parameter is , for various system sizes, in complete agreement with
the experimental results. However, the fluctuations of do not show any
critical behavior, probably due to crossover effects by the close presence of
the tricritical point. Finally, in quasi-one-dimensional systems, the
transition is only discontinuous, limited by one critical point, indicating
that two is the lower dimension for having a tricritical point
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