Showing papers on "Correlation function (statistical mechanics) published in 2016"

Observation of Noise Correlated by the Hawking Effect in a Water Tank

Abstract: We measured the power spectrum and two-point correlation function for the randomly fluctuating free surface on the downstream side of a stationary flow with a maximum Froude number ${F}_{\mathrm{max}}\ensuremath{\approx}0.85$ reached above a localized obstacle. On such a flow the scattering of incident long wavelength modes is analogous to that responsible for black hole radiation (the Hawking effect). Our measurements of the noise show a clear correlation between pairs of modes of opposite energies. We also measure the scattering coefficients by applying the same analysis of correlations to waves produced by a wave maker.

Fast large scale structure perturbation theory using one-dimensional fast Fourier transforms

Abstract: Author(s): Schmittfull, M; Vlah, Z; McDonald, P | Abstract: The usual fluid equations describing the large-scale evolution of mass density in the universe can be written as local in the density, velocity divergence, and velocity potential fields. As a result, the perturbative expansion in small density fluctuations, usually written in terms of convolutions in Fourier space, can be written as a series of products of these fields evaluated at the same location in configuration space. Based on this, we establish a new method to numerically evaluate the 1-loop power spectrum (i.e.; Fourier transform of the 2-point correlation function) with one-dimensional fast Fourier transforms. This is exact and a few orders of magnitude faster than previously used numerical approaches. Numerical results of the new method are in excellent agreement with the standard quadrature integration method. This fast model evaluation can in principle be extended to higher loop order where existing codes become painfully slow. Our approach follows by writing higher order corrections to the 2-point correlation function as, e.g.; the correlation between two second-order fields or the correlation between a linear and a third-order field. These are then decomposed into products of correlations of linear fields and derivatives of linear fields. The method can also be viewed as evaluating three-dimensional Fourier space convolutions using products in configuration space, which may also be useful in other contexts where similar integrals appear.

Micro-heterogeneity versus clustering in binary mixtures of ethanol with water or alkanes

Abstract: Ethanol is a hydrogen bonding liquid. When mixed in small concentrations with water or alkanes, it forms aggregate structures reminiscent of, respectively, the direct and inverse micellar aggregates found in emulsions, albeit at much smaller sizes. At higher concentrations, micro-heterogeneous mixing with segregated domains is found. We examine how different statistical methods, namely correlation function analysis, structure factor analysis and cluster distribution analysis, can describe efficiently these morphological changes in these mixtures. In particular, we explain how the neat alcohol pre-peak of the structure factor evolves into the domain pre-peak under mixing conditions, and how this evolution differs whether the co-solvent is water or alkane. This study clearly establishes the heuristic superiority of the correlation function/structure factor analysis to study the micro-heterogeneity, since cluster distribution analysis is insensitive to domain segregation. Correlation functions detect the domains, with a clear structure factor pre-peak signature, while the cluster techniques detect the cluster hierarchy within domains. The main conclusion is that, in micro-segregated mixtures, the domain structure is a more fundamental statistical entity than the underlying cluster structures. These findings could help better understand comparatively the radiation scattering experiments, which are sensitive to domains, versus the spectroscopy-NMR experiments, which are sensitive to clusters.

Accelerating the two-point and three-point galaxy correlation functions using Fourier transforms

Abstract: Though Fourier Transforms (FTs) are a common technique for finding correlation functions, they are not typically used in computations of the anisotropy of the two-point correlation function (2PCF) about the line of sight in wide-angle surveys because the line-of-sight direction is not constant on the Cartesian grid. Here we show how FTs can be used to compute the multipole moments of the anisotropic 2PCF. We also show how FTs can be used to accelerate the 3PCF algorithm of Slepian & Eisenstein (2015). In both cases, these FT methods allow one to avoid the computational cost of pair counting, which scales as the square of the number density of objects in the survey. With the upcoming large datasets of DESI, Euclid, and LSST, FT techniques will therefore offer an important complement to simple pair or triplet counts.

Improving the modelling of redshift-space distortions - II. A pairwise velocity model covering large and small scales

Abstract: We develop a model for the redshift-space correlation function, valid for both dark matter particles and halos on scales $>5\,h^{-1}$Mpc. In its simplest formulation, the model requires the knowledge of the first three moments of the line-of-sight pairwise velocity distribution plus two well-defined dimensionless parameters. The model is obtained by extending the Gaussian-Gaussianity prescription for the velocity distribution, developed in a previous paper, to a more general concept allowing for local skewness, which is required to match simulations. We compare the model with the well known Gaussian streaming model and the more recent Edgeworth streaming model. Using N-body simulations as a reference, we show that our model gives a precise description of the redshift-space clustering over a wider range of scales. We do not discuss the theoretical prescription for the evaluation of the velocity moments, leaving this topic to further investigation.

Long-term study of Mkn 421 with the HAGAR array of telescopes

Abstract: Context. The HAGAR Telescope Array at Hanle, Ladakh has been regularly monitoring the nearby blazar Mkn 421 for the past seven years.Aims. Blazars show flux variability in all timescales across the electromagnetic spectrum. While there is abundant literature characterizing the short-term flares from different blazars, comparatively little work has been carried out to study the long-term variability. We aim to study the long-term temporal and spectral variability in the radiation from Mkn 421 during 2009−2015.Methods. We quantify the variability and lognormality from the radio to very high-energy (VHE) bands, and compute the correlations between various wavebands using the z -transformed discrete correlation function. We construct the spectral energy distribution (SED) contemporaneous with HAGAR observation seasons and fit this SED with a one-zone synchrotron self-Compton model to study the spectral variability.Results. The flux is found to be highly variable across all timescales. The variability is energy dependant and is maximum in the X-ray and VHE bands. A strong correlation is found between the Fermi -LAT (gamma) and radio bands and between Fermi -LAT and optical, but no correlation is found between Fermi -LAT and X-ray. Lognormality in the flux distribution is clearly detected. This is the third blazar, following BL Lac and PKS 2155+304 to show this behaviour. The SED can be well fit by a one-zone SSC model, and variations in the flux states can be attributed mainly to changes in the particle distribution. A strong correlation is seen between the break energy γ b of the particle spectrum and the total bolometric luminosity.

3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise

Abstract: We consider a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4.

Momentum-Space Correlations of a One-Dimensional Bose Gas.

Abstract: Analyzing the noise in the momentum profiles of single realizations of one-dimensional Bose gases, we present the experimental measurement of the full momentum-space density correlations ⟨δn_{p}δn_{p^{'}}⟩, which are related to the two-body momentum correlation function. Our data span the weakly interacting region of the phase diagram, going from the ideal Bose gas regime to the quasicondensate regime. We show experimentally that the bunching phenomenon, which manifests itself as super-Poissonian local fluctuations in momentum space, is present in all regimes. The quasicondensate regime is, however, characterized by the presence of negative correlations between different momenta, in contrast to the Bogolyubov theory for Bose condensates, predicting positive correlations between opposite momenta. Our data are in good agreement with ab initio calculations.

A multiple length scale correlation operator for ocean data assimilation

Abstract: Ocean data assimilation systems can take into account time and space scale variations by representing background error covariance functions with more complex shapes than the classical Gaussian function. In particular, the construction of the correlation functions can be improved to give more flexibility. We describe a correlation operator that features high correlations within a short scale and weak correlations within a larger scale. This multiple length scale correlation operator is defined as a linear combination of Whittle–Matern functions with different length scales. The main characteristics of the resulting correlation function are described. In particular, a focus is given on features that might be of interest to determine the parameters of the model: the Daley length scale, the normalised spectrum inflexion point and the kurtosis coefficient. The multiple length scale operator has been implemented in NEMOVAR, a variational ocean data assimilation system. A dual length scale formulation was tested in a one-year reanalysis and compared with a single length scale formulation. The results emphasise the importance of estimating with great care the factors used within the combination. They also demonstrate the potential of the dual length scale formulation, in particular through a decrease of the innovation statistics for salinity profiles. The dual length scale formulation is now operational at the Met Office. Keywords: background error, covariance, correlations, length scales, diffusion equation, recursive filter, kurtosis (Published: 25 February 2016) Citation: Tellus A 2016, 68, 29744, http://dx.doi.org/10.3402/tellusa.v68.29744

Estimation and optimization of loss-of-pair uncertainties based on PIV correlation functions

Abstract: The uncertainty quantification of particle image velocimetry (PIV) measurements is still an open problem, and to date, no consensus exists about the best suited approach. When the spatial resolution is not appropriate, the largest uncertainties are usually caused by flow gradients. But also the amount of loss-of-pairs due to out-of-plane flow motion and insufficient light-sheet overlap causes strong uncertainties in real experiments. In this paper, we show how the amount of loss-of-pairs can be quantified using the volume of the correlation function normalized by the volume of the autocorrelation function. The findings are an important step toward a reliable uncertainty estimation of instantaneous planar velocity fields computed from PIV and stereo-PIV data. Another important consequence of the analysis is that the results allow for the optimization of PIV and stereo-PIV setups in view of minimizing the total error. In particular, it is shown that the best results (concerning the relative uncertainty) can be achieved if the out-of-plane loss-of-correlation is smaller than one (F o ). The only exception is the case where the out-of-plane motion is exactly zero. The predictions are confirmed experimentally in the last part of the paper.

Correlation and fluctuation in a random average process on an infinite line with a driven tracer

Abstract: We study the effect of a single biased tracer particle in a bath of other particles performing the random average process (RAP) on an infinite line. We focus on the long time behavior of the mean and the fluctuations of the positions of the particles and also the correlations among them. In the long time t limit these quantities have well defined scaling forms and grow with time as . A differential equation for the scaling function associated with the correlation function is obtained and solved perturbatively around the solution for a symmetric tracer. Interestingly, when the tracer is totally asymmetric, further progress is enabled by the fact that the particles behind the tracer do not affect the motion of the particles in front of it, which leads in particular to an exact expression for the variance of the position of the tracer. Finally, the variance and correlations of the gaps between successive particles are also studied. Numerical simulations support our analytical results.

A Novel Wideband MIMO Car-to-Car Channel Model Based on a Geometrical Semi-Circular Tunnel Scattering Model

Abstract: In this paper, we present a wideband multiple-input multiple-output (MIMO) car-to-car (C2C) channel model based on a geometrical semi-circular tunnel (SCT) scattering model. From the geometrical SCT scattering model, a reference channel model is derived under the assumption of single-bounce scattering in line-of-sight (LOS) and non-LOS (NLOS) propagation environments. In the proposed reference channel model, it is assumed that an infinite number of scatterers are randomly distributed on the tunnel wall. Starting from the geometrical scattering model, the time-variant transfer function (TVTF) is derived, and its correlation properties in time, frequency, and space are studied. Expressions are presented for the space–time–frequency cross-correlation function (STF-CCF), the two-dimensional (2D) space CCF, the 2D time–frequency CCF (TF-CCF), the temporal autocorrelation function (ACF), and the frequency correlation function (FCF). Owing to the semi-circular geometry, we reduced the originally threefold integrals to double integrals in the computations of the correlation functions, which simplifies the numerical analysis considerably. From the TVTF characterizing the reference model, an efficient sum-of-cisoid (SOC) channel simulator is derived. Numerical results show that both the temporal ACF and the FCF of the SOC channel simulator match very well with those of the reference model. A validation of the proposed model has been done by fitting the delay spread of the reference model to that of the measured channel, which demonstrates an excellent agreement. The proposed channel simulator allows us to evaluate the performance of C2C communication systems in tunnel environments.

Hydrogen bonded clusters in the liquid phase: I. Analysis of the velocity correlation function of

Abstract: We present an analysis of hydrogen bonded triplet clusters of water in terms of velocity projections on locally defined molecular coordinates. This makes it possible to perform an approximate normal mode analysis and to establish the relative motions of hydrogen bonded molecules. We find that the interpretation of the velocity autocorrelation function (VACF) of water in terms of stretching and bending motions of hydrogen bonded neighbours is an oversimplification due to cooperative effects in the hydrogen bonded network. Especially, for the peak at we find that it is not only due to O-O-O bending motions in hydrogen bonded groups but also to librational motions of the whole cluster.

Ensemble Space-Time Correlation of Plasma Turbulence in the Solar Wind.

Abstract: Single point measurement turbulence cannot distinguish variations in space and time. We employ an ensemble of one- and two-point measurements in the solar wind to estimate the space-time correlation function in the comoving plasma frame. The method is illustrated using near Earth spacecraft observations, employing ACE, Geotail, IMP-8, and Wind data sets. New results include an evaluation of both correlation time and correlation length from a single method, and a new assessment of the accuracy of the familiar frozen-in flow approximation. This novel view of the space-time structure of turbulence may prove essential in exploratory space missions such as Solar Probe Plus and Solar Orbiter for which the frozen-in flow hypothesis may not be a useful approximation.

Effective Theory of Squeezed Correlation Functions

Abstract: Various inflationary scenarios can often be distinguished from one another by looking at the squeezed limit behavior of correlation functions. Therefore, it is useful to have a framework designed to study this limit in a more systematic and efficient way. We propose using an expansion in terms of weakly coupled super-horizon degrees of freedom, which is argued to generically exist in a near de Sitter space-time. The modes have a simple factorized form which leads to factorization of the squeezed-limit correlation functions with power-law behavior in klong/kshort. This approach reproduces the known results in single-, quasi-single-, and multi-field inflationary models. However, it is applicable even if, unlike the above examples, the additional degrees of freedom are not weakly coupled at sub-horizon scales. Stronger results are derived in two-field (or sufficiently symmetric multi-field) inflationary models. We discuss the observability of the non-Gaussian 3-point function in the large-scale structure surveys, and argue that the squeezed limit behavior has a higher detectability chance than equilateral behavior when it scales as (klong/kshort)Δ with Δ < 1—where local non-Gaussianity corresponds to Δ = 0.

Prethermalization and thermalization of a quenched interacting Luttinger liquid

Abstract: We study the relaxation dynamics of interacting one-dimensional fermions with band curvature after a weak quench in the interaction parameter at zero temperature. Our model lies within the class of interacting Luttinger liquids, where the harmonic Luttinger theory is extended by a weak-integrability-breaking phonon scattering term. In order to solve for the nonequilibrium time evolution, we use quantum kinetic equations exploiting the resonant but subleading character of the phonon interaction term. The interplay between phonon scattering and the quadratic Luttinger theory leads to the emergence of three distinct spatiotemporal regimes for the fermionic real-space correlation function. It features the crossover from a prequench to a prethermal state, finally evolving towards a thermal state on increasing length and time scales. The characteristic algebraically decaying real-space correlations in the prethermalized regime become modulated by an amplitude that is decaying in time according to a stretched exponential as an effect of the interactions. The asymptotic thermalization dynamics is governed by energy transport over large distances from the thermalized to the nonthermalized regions via macroscopic, dynamical slow modes. This is revealed in an algebraic decay of the system's effective temperature. The numerical value of the associated exponent agrees with the dynamical critical exponent of the Kardar-Parisi-Zhang universality class. We also discuss a criterion for the applicability of this theory away from the integrable limit of noninteracting fermions.

Relating optical and microwave grain metrics of snow: the relevance of grain shape

Abstract: . Grain shape is commonly understood as a morphological characteristic of snow that is independent of the optical diameter (or specific surface area) influencing its physical properties. In this study we use tomography images to investigate two objectively defined metrics of grain shape that naturally extend the characterization of snow in terms of the optical diameter. One is the curvature length λ2, related to the third-order term in the expansion of the two-point correlation function, and the other is the second moment μ2 of the chord length distributions. We show that the exponential correlation length, widely used for microwave modeling, can be related to the optical diameter and λ2. Likewise, we show that the absorption enhancement parameter B and the asymmetry factor gG, required for optical modeling, can be related to the optical diameter and μ2. We establish various statistical relations between all size metrics obtained from the two-point correlation function and the chord length distribution. Overall our results suggest that the characterization of grain shape via λ2 or μ2 is virtually equivalent since both capture similar aspects of size dispersity. Our results provide a common ground for the different grain metrics required for optical and microwave modeling of snow.

3D conformation of a flexible fiber in a turbulent flow

Abstract: A growing number of studies is devoted to anisotropic particles in turbulent flows. In most cases the particles are assumed to be rigid and their deformations are neglected. We present an adaptation of classical computer vision tools to reconstruct from two different images the 3D conformation of a fiber distorted by the turbulent fluctuations in a von Karman flow. This technique allows us notably to characterize the fiber deformation by computing the correlation function of the orientation of the tangent vector. This function allows us to tackle the analogy between polymers and flexible fibers proposed by Brouzet et al. (Phys. Rev. Lett. 112(7) 074501 (2014)). We show that this function depends on an elastic length e which characterizes the particle flexibility, as is the case for polymers, but also on the fiber length L, contrary to polymers.

Baseline-free fatigue crack detection based on spectral correlation and nonlinear wave modulation

Abstract: By generating ultrasonic waves at two different frequencies onto a cracked structure, modulations due to crack-induced nonlinearity can be observed in the corresponding ultrasonic response. This nonlinear wave modulation phenomenon has been widely studied and proven capable of detecting a fatigue crack at a very early stage. However, under field conditions, other exogenous vibrations exist and the modulation components can be buried under ambient noises, making it difficult to extract the modulation components simply by using a spectral density function. In this study, the nonlinear modulation components in the ultrasonic response were extracted using a spectral correlation function (the double Fourier transform) with respect to time and time lag of a signal's autocorrelation. Using spectral correlation, noise or interference, which is spectrally overlapped with the nonlinear modulation components in the ultrasonic response, can be effectively removed or reduced. Only the nonlinear modulation components are accentuated at specific coordinates of the spectral correlation plot. A damage feature is defined by comparing the spectral correlation value between nonlinear modulation components with other spectral correlation values among randomly selected frequencies. Then, by analyzing the statistical characteristics of the multiple damage feature values obtained from different input frequency combinations, fatigue cracks can be detected without relying on baseline data obtained from the pristine condition of the target structure. In the end, an experimental test was conducted on aluminum plates with a real fatigue crack and the test signals were contaminated by simulated noises with varying signal-to-noise ratios. The results validated the proposed technique.

Correlation and fluctuation in Random Average Process on an infinite line with a driven tracer

Abstract: We study the effect of single biased tracer particle in a bath of other particles performing the random average process (RAP) on an infinite line. We focus on the large time behavior of the mean and the fluctuations of the positions of the particles and also the correlations among them. In the large time t limit these quantities have well-defined scaling forms and grow with time as $\sqrt{t}$. A differential equation for the scaling function associated with the correlation function is obtained and solved perturbatively around the solution for a symmetric tracer. Interestingly, when the tracer is totally asymmetric, further progress is enabled by the fact that the particles behind of the tracer do not affect the motion of the particles in front of it, which leads in particular to an exact expression for the variance of the position of the tracer. Finally, the variance and correlations of the gaps between successive particles are also studied. Numerical simulations support our analytical results.

Communication: Temperature derivative of the dielectric constant gives access to multipoint correlations in polar liquids

Abstract: Fluctuations of the dipole moment of a macroscopic dielectric sample are induced by thermal motions. The variance of the sample dipole moment, characterizing the extent of thermal fluctuations, is a decaying function of temperature for many polar liquids. This result is inconsistent with the Nyquist (fluctuation-dissipation) theorem predicting the variance of a macroscopic property to grow linearly with temperature. The reason for a qualitatively different behavior is in strong multi-particle correlations of dipolar orientations. An equation connecting the temperature slope of the dielectric constant to a static three-point correlation function is derived. When applied to experimental data for polar and hydrogen-bonding liquids at normal conditions, the three-point correlations of different liquids fall on a single master curve as a function of the dielectric constant. Static three-point correlation functions can potentially reflect the growing spatial correlation length on approach to the glass transition. However, the measured temperature slope of the dielectric constant of glycerol does not indicate a change in such a lengthscale.

Microscopic treatment of solute trapping and drag.

Abstract: The long wavelength limit of a recent microscopic phase-field crystal (PFC) theory of a binary alloy mixture is used to derive an analytical approximation for the segregation coefficient as a function of the interface velocity, and relate it to the two-point correlation function of the liquid and the thermodynamic properties of solid and liquid phases. Our results offer the first analytical derivation of solute segregation from a microscopic model, and support recent molecular dynamics and numerical PFC simulations. Our results also provide an independent framework, motivated from classical density functional theory, from which to elucidate the fundamental nature of solute drag, which is still highly contested in the literature.

On the spatio-temporal behavior of magnetohydrodynamic turbulence in a magnetized plasma

Abstract: Using direct numerical simulations of three-dimensional magnetohydrodynamic turbulence, the spatio-temporal behavior of magnetic field fluctuations is analyzed. Cases with relatively small, medium, and large values of a mean background magnetic field are considered. The (wavenumber) scale dependent time correlation function is directly computed for different simulations, varying the mean magnetic field value. From this correlation function, the time decorrelation is computed and compared with different theoretical times, namely, the local non-linear time, the random sweeping time, and the Alfvenic time, the latter being a wave effect. It is observed that time decorrelations are dominated by sweeping effects, and only at large values of the mean magnetic field and for wave vectors mainly aligned with this field time decorrelations are controlled by Alfvenic effects.

Tuning Pairwise Potential Can Control the Fragility of Glass-Forming Liquids: From Tetrahedral Network to Isotropic Soft Sphere Models

Abstract: We perform molecular dynamics simulations for a silica glass former model proposed by Coslovich and Pastore (CP) over a wide range of densities. The density variation can be mapped onto the change of the potential depth between Si and O interactions of the CP model. By reducing the potential depth (or increasing the density), the anisotropic tetrahedral network structure observed in the original CP model transforms into the isotropic structure with the purely repulsive soft-sphere potential. Correspondingly, the temperature dependence of the relaxation time exhibits the crossover from the Arrhenius to the super-Arrhenius behavior. Being able to control the fragility over a wide range by tuning the potential of a single model system helps us to bridge the gap between the network and isotropic glass formers and to obtain the insight into the underlying mechanism of the fragility. We study the relationship between the fragility and dynamical properties such as the magnitude of the Stokes-Einstein violation and the stretch exponent in the density correlation function. We also demonstrate that the peak of the specific heat systematically shifts as the density increases, hinting that the fragility is correlated with the hidden thermodynamic anomalies of the system.

Higher order correlation beams in atmosphere under strong turbulence conditions.

Abstract: Higher order correlation beams, that is, two-photon beams obtained from the process of spontaneous parametric down-conversion pumped by Hermite-Gauss or Laguerre-Gauss beams of any order, can be used to encode information in many modes, opening the possibility of quantum communication with large alphabets. In this paper we calculate, analytically, the fourth-order correlation function for the Hermite-Gauss and Laguerre-Gauss coherent and partially coherent correlation beams propagating through a strong turbulent medium. We show that fourth-order correlation functions for correlation beams have, under certain conditions, expressions similar to those of intensities of classical beams and are degraded by turbulence in a similar way as the classical beams. Our results can be useful in establishing limits for the use of two-photon beams in quantum communications with larger alphabets under atmospheric turbulence.

Direct Computation of Influence Functional Coefficients from Numerical Correlation Functions

Abstract: Influence functional methods provide a powerful approach for simulating the dynamics of a system embedded in a harmonic bath, which may be parametrized to reflect a variety of environments and chemical processes. In this work, we develop a procedure for calculating the coefficients of the discretized influence functional using the classical approximation to the time correlation function, which is usually available through molecular dynamics simulations. This procedure circumvents the calculation of a spectral density via Fourier inversion of the correlation function. When the correlation function is available with high precision, we find that the direct procedure yields results just as accurate as those obtained using the spectral density expressions. However, when statistical noise is present, the direct procedure produces more accurate results. The direct procedure is efficient and easy to implement.

The Dynamic Correlation Model and Its Application to the Heston Model

Abstract: Correlation plays an essential role in many problems of finance and economics, such as pricing financial products and hedging strategies, since it models the degree of relationship between, e.g., financial products and financial institutions. However, usually for simplicity the correlation coefficient is assumed to be a constant in many models, although financial quantities are correlated in a strongly nonlinear way in the real market. This work provides a new time-dependent correlation function, which can be easily used to construct dynamically (time-dependent) correlated Brownian motions and flexibly incorporated in many financial models. The aim of using our time-dependent correlation function is to reasonably choose additional parameters to increase the fitting quality on the one hand, but also add an economic concept on the other hand. As examples, we illustrate the applications of dynamic correlation in the Heston model. From our numerical results we conclude that the Heston model extended by incorporating time-dependent correlations can provide a better volatility smile than the pure Heston model.