Second order analysis of geometric functionals of Boolean models

This paper presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values.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 Jensen. (The second version mainly resolves minor LaTeX problems.

Microscopic theory for interface fluctuations in binary liquid mixtures

Thermally excited capillary waves at fluid interfaces in binary liquid mixtures exhibit simultaneously both density and composition fluctuations. Based on a density functional theory for inhomogeneous binary liquid mixtures we derive an effective wavelength dependent Hamiltonian for fluid interfaces in these systems beyond the standard capillary-wave model. Explicit expressions are obtained for the surface tension, the bending rigidities, and the coupling constants of compositional capillary waves in terms of the profiles of the two number densities characterizing the mixture. These results lead to predictions for grazing-incidence x-ray scattering experiments at such interfaces.Comment: 23 pages, 11 figure

Local orientations of fluctuating fluid interfaces

Thermal fluctuations cause the local normal vectors of fluid interfaces to deviate from the vertical direction defined by the flat mean interface position. This leads to a nonzero mean value of the corresponding polar tilt angle which renders a characterization of the thermal state of an interface. Based on the concept of an effective interface Hamiltonian we determine the variances of the local interface position and of its lateral derivatives. This leads to the probability distribution functions for the metric of the interface and for the tilt angle which allows us to calculate its mean value and its mean square deviation. We compare the temperature dependences of these quantities as predicted by the simple capillary wave model, by an improved phenomenological model, and by the microscopic effective interface Hamiltonian derived from density functional theory. The mean tilt angle discriminates clearly between these theoretical approaches and emphasizes the importance of the variation of the surface tension at small wave lengths. Also the tilt angle two-point correlation function is determined which renders an additional structural characterization of interfacial fluctuations. Various experimental accesses to measure the local orientational fluctuations are discussed.Comment: 29 pages, 12 figure

Radial Distribution Function for Semiflexible Polymers Confined in Microchannels

An analytic expression is derived for the distribution $G(\vec{R})$ of the end-to-end distance $\vec{R}$ of semiflexible polymers in external potentials to elucidate the effect of confinement on the mechanical and statistical properties of biomolecules. For parabolic confinement the result is exact whereas for realistic potentials a self-consistent ansatz is developed, so that $G(\vec{R})$ is given explicitly even for hard wall confinement. The theoretical result is in excellent quantitative agreement with fluorescence microscopy data for actin filaments confined in rectangularly shaped microchannels. This allows an unambiguous determination of persistence length $L_P$ and the dependence of statistical properties such as Odijk's deflection length $\lambda$ on the channel width $D$. It is shown that neglecting the effect of confinement leads to a significant overestimation of bending rigidities for filaments

Percolation Analysis of a Wiener Reconstruction of the IRAS 1.2 Jy Redshift Catalog

We present percolation analyses of Wiener Reconstructions of the IRAS 1.2 Jy Redshift Survey. There are ten reconstructions of galaxy density fields in real space spanning the range $\beta= 0.1$ to $1.0$, where ${\beta}={\Omega^{0.6}}/b$, $\Omega$ is the present dimensionless density and $b$ is the bias factor. Our method uses the growth of the largest cluster statistic to characterize the topology of a density field, where Gaussian randomized versions of the reconstructions are used as standards for analysis. For the reconstruction volume of radius, $R {\approx} 100 h^{-1}$ Mpc, percolation analysis reveals a slight `meatball' topology for the real space, galaxy distribution of the IRAS survey. cosmology-galaxies:clustering-methods:numericalComment: Revised version accepted for publication in The Astrophysical Journal, January 10, 1997 issue, Vol.47

Using the filaments in the LCRS to test the LambdaCDM model

It has recently been established that the filaments seen in the Las Campanas Redshift Survey (LCRS) are statistically significant at scales as large as 70 to 80 Mpc/h in the $-3^{\circ}$ slice, and 50 to 70 Mpc/h in the five other LCRS slices. The ability to produce such filamentary features is an important test of any model for structure formation. We have tested the LCDM model with a featureless, scale invariant primordial power spectrum by quantitatively comparing the filamentarity in simulated LCRS slices with the actual data. The filamentarity in an unbiased LCDM model, we find, is less than the LCRS. Introducing a bias b=1.15, the model is in rough consistency with the data, though in two of the slices the filamentarity falls below the data at a low level of statistical significance. The filamentarity is very sensitive to the bias parameter and a high value b=1.5, which enhances filamentarity at small scales and suppresses it at large scales, is ruled out. A bump in the power spectrum at k~0.05 Mpc/h is found to have no noticeable effect on the filamentarity.Comment: 16 pages, 3 figures; Minor Changes, Accepted to Ap

Evidence for Filamentarity in the Las Campanas Redshift Survey

We apply Shapefinders, statistical measures of `shape' constructed from two dimensional partial Minkowski functionals, to study the degree of filamentarity in the Las Campanas Redshift Survey (LCRS). In two dimensions, three Minkowski functionals characterise the morphology of an object, they are: its perimeter (L), area (S), and genus. Out of L and S a single dimensionless Shapefinder Statistic, F can be constructed (0 <=F <=1). F acquires extreme values on a circle (F = 0) and a filament (F = 1). Using F, we quantify the extent of filamentarity in the LCRS by comparing our results with a Poisson distribution with similar geometrical properties and having the same selection function as the survey. Our results unambiguously demonstrate that the LCRS displays a high degree of filamentarity both in the Northern and Southern galactic sections a result that is in general agreement with the visual appearance of the catalogue. It is well known that gravitational clustering from Gaussian initial conditions gives rise to the development of non-Gaussianity reflected in the formation of a network-like filamentary structure on supercluster scales. Consequently the fact that the smoothed LCRS catalogue shows properties consistent with those of a Gaussian random field (Colley 1997) whereas the unsmoothed catalogue demonstrates the presence of filamentarity lends strong support to the conjecture that the large scale clustering of galaxies is driven by gravitational instability.Comment: Accepted for publication in Ap

Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

Effect of an electric field on a floating lipid bilayer: a neutron reflectivity study

We present here a neutron reflectivity study of the influence of an alternative electric field on a supported phospholipid double bilayer. We report for the first time a reproducible increase of the fluctuation amplitude leading to the complete unbinding of the floating bilayer. Results are in good agreement with a semi-quantitative interpretation in terms of negative electrostatic surface tension.Comment: 12 pages, 7 figures, 1 table accepted for publication in European Physical Journal E Replaced with with correct bibliograph