11,488 research outputs found
Phase separation dynamics of polydisperse colloids: a mean-field lattice-gas theory
New insights into phase separation in colloidal suspensions are provided via
a new dynamical theory based on the Polydisperse Lattice-Gas model. The model
gives a simplified description of polydisperse colloids, incorporating a
hard-core repulsion combined with polydispersity in the strength of the
attraction between neighbouring particles. Our mean-field equations describe
the local concentration evolution for each of an arbitrary number of species,
and for an arbitrary overall composition of the system. We focus on the
predictions for the dynamics of colloidal gas-liquid phase separation after a
quench into the coexistence region. The critical point and the relevant
spinodal curves are determined analytically, with the latter depending only on
three moments of the overall composition. The results for the early-time
spinodal dynamics show qualitative changes as one crosses a 'quenched' spinodal
that excludes fractionation and so allows only density fluctuations at fixed
composition. This effect occurs for dense systems, in agreement with a
conjecture by Warren that, at high density, fractionation should be generically
slow because it requires inter-diffusion of particles. We verify this
conclusion by showing that the observed qualitative changes disappear when
direct particle-particle swaps are allowed in the dynamics. Finally, the rich
behaviour beyond the spinodal regime is examined, where we find that the
evaporation of gas bubbles with strongly fractionated interfaces causes
long-lived composition heterogeneities in the liquid phase; we introduce a
two-dimensional density histogram method that allows such effects to be easily
visualized for an arbitrary number of particle species.Comment: 20 pages; accepted for publication in Physical Chemistry Chemical
Physic
Principal Component Analysis as a Tool for Characterizing Black Hole Images and Variability
We explore the use of principal component analysis (PCA) to characterize
high-fidelity simulations and interferometric observations of the millimeter
emission that originates near the horizons of accreting black holes. We show
mathematically that the Fourier transforms of eigenimages derived from PCA
applied to an ensemble of images in the spatial-domain are identical to the
eigenvectors of PCA applied to the ensemble of the Fourier transforms of the
images, which suggests that this approach may be applied to modeling the sparse
interferometric Fourier-visibilities produced by an array such as the Event
Horizon Telescope (EHT). We also show that the simulations in the spatial
domain themselves can be compactly represented with a PCA-derived basis of
eigenimages allowing for detailed comparisons between variable observations and
time-dependent models, as well as for detection of outliers or rare events
within a time series of images. Furthermore, we demonstrate that the spectrum
of PCA eigenvalues is a diagnostic of the power spectrum of the structure and,
hence, of the underlying physical processes in the simulated and observed
images.Comment: 16 pages, 17 figures, submitted to Ap
Bridge trisections of knotted surfaces in 4--manifolds
We prove that every smoothly embedded surface in a 4--manifold can be
isotoped to be in bridge position with respect to a given trisection of the
ambient 4--manifold; that is, after isotopy, the surface meets components of
the trisection in trivial disks or arcs. Such a decomposition, which we call a
\emph{generalized bridge trisection}, extends the authors' definition of bridge
trisections for surfaces in . Using this new construction, we give
diagrammatic representations called \emph{shadow diagrams} for knotted surfaces
in 4--manifolds. We also provide a low-complexity classification for these
structures and describe several examples, including the important case of
complex curves inside . Using these examples, we prove that
there exist exotic 4--manifolds with --trisections for certain values of
. We conclude by sketching a conjectural uniqueness result that would
provide a complete diagrammatic calculus for studying knotted surfaces through
their shadow diagrams.Comment: 17 pages, 5 figures. Comments welcom
Isotropic-nematic phase equilibria of polydisperse hard rods: The effect of fat tails in the length distribution
We study the phase behaviour of hard rods with length polydispersity, treated
within a simplified version of the Onsager model. We give a detailed
description of the unusual phase behaviour of the system when the rod length
distribution has a "fat" (e.g. log-normal) tail up to some finite cutoff. The
relatively large number of long rods in the system strongly influences the
phase behaviour: the isotropic cloud curve, which defines the where a nematic
phase first occurs as density is increased, exhibits a kink; at this point the
properties of the coexisting nematic shadow phase change discontinuously. A
narrow three-phase isotropic-nematic-nematic coexistence region exists near the
kink in the cloud curve, even though the length distribution is unimodal. A
theoretical derivation of the isotropic cloud curve and nematic shadow curve,
in the limit of large cutoff, is also given. The two curves are shown to
collapse onto each other in the limit. The coexisting isotropic and nematic
phases are essentially identical, the only difference being that the nematic
contains a larger number of the longest rods; the longer rods are also the only
ones that show any significant nematic ordering. Numerical results for finite
but large cutoff support the theoretical predictions for the asymptotic scaling
of all quantities with the cutoff length.Comment: 21 pages, 13 figure
Productivity Measurement with Nonstatic Expectations and Varying Capacity Utilization: An Integrated Approach
Typically measures of multifactor productivity growth have been based on a production and optimization framework that assumes all inputs are instantaneously adjustable, thus ignoring the important impacts of short run fixity of certain inputs. This paper focuses on the distinction between short and long run production behavior represented by economic capacity utilization indexes, and on the adjustment of observed productivity measures for the effects of short run fixity characterized by these indexes. A dynamic optimization model based on adjustment costs for quasi-fixed inputs is developed to calculate capacity utilization adjustments for productivity growth measures. The resulting framework is then used to identify empirically the effects of capacity utilization, nonstatic expectations,nonconstant returns to scale and adjustment costs for both capital and labor on productivity growth in the U.S. manufacturing sector, 1947-1979.
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