Showing papers on "Correlation function (statistical mechanics) published in 1999"

Dynamic density functional theory of fluids

Abstract: We present a new time-dependent density functional approach to study the relaxational dynamics of an assembly of interacting particles subject to thermal noise. Starting from the Langevin stochastic equations of motion for the velocities of the particles we are able by means of an approximated closure to derive a self-consistent deterministic equation for the temporal evolution of the average particle density. The closure is equivalent to assuming that the equal-time two-point correlation function out of equilibrium has the same properties as its equilibrium version. The changes in time of the density depend on the functional derivatives of the grand canonical free energy functional F[ρ] of the system. In particular the static solutions of the equation for the density correspond to the exact equilibrium profiles provided one is able to determine the exact form of F[ρ]. In order to assess the validity of our approach we performed a comparison between the Langevin dynamics and the dynamic density functional...

On the relevance of long-range dependence in network traffic

Abstract: There is much experimental evidence that network traffic processes exhibit ubiquitous properties of self-similarity and long-range dependence, i.e., of correlations over a wide range of time scales. However, there is still considerable debate about how to model such processes and about their impact on network and application performance. In this paper, we argue that much previous modeling work has failed to consider the impact of two important parameters, namely the finite range of time scales of interest in performance evaluation and prediction problems, and the first-order statistics such as the marginal distribution of the process. We introduce and evaluate a model in which these parameters can be controlled. Specifically, our model is a modulated fluid traffic model in which the correlation function of the fluid rate matches that of an asymptotically second-order self-similar process with given Hurst parameter up to an arbitrary cutoff time lag, then drops to zero. We develop a very efficient numerical procedure to evaluate the performance of a single-server queue fed with the above fluid input process. We use this procedure to examine the fluid loss rate for a wide range of marginal distributions, Hurst (1950) parameters, cutoff lags, and buffer sizes. Our main results are as follows. First, we find that the amount of correlation that needs to be taken into account for performance evaluation depends not only on the correlation structure of the source traffic, but also on time scales specific to the system under study. For example, the time scale associated with a queueing system is a function of the maximum buffer size. Thus, for finite buffer queues, we find that the impact on loss of the correlation in the arrival process becomes nil beyond a time scale we refer to as the correlation horizon. This means, in particular, that for performance-modeling purposes, we may choose any model among the panoply of available models (including Markovian and self-similar models) as long as the chosen model captures the correlation structure of the source traffic up to the correlation horizon. Second, we find that loss can depend in a crucial way on the marginal distribution of the fluid rate process. Third, our results suggest that reducing loss by buffering is hard for traffic with correlation over many time scales. We advocate the use of source traffic control and statistical multiplexing instead.

Dynamics of axisymmetric and poloidal flows in tokamaks

Abstract: As a result of turbulence and finite Larmor radius effects, random radial currents are present in a tokamak plasma and these drive sheared axisymmetric poloidal flows. We model these currents with a noise source with given statistical properties and calculate the linear kinetic response to this source. Without collisions, there is no long term damping of these flows; when collisions are included, poloidal flows are damped. The mean square potential associated with these flows is given in terms of the linear response function we calculate and a model correlation function for the current source. Without collisions, the mean square flow increases linearly with time, but with collisions, it reaches a steady state. In the long correlation time limit, the collisionless residual flows are important in determining the mean square flow.

Weak Lensing with SDSS Commissioning Data: The Galaxy-Mass Correlation Function To 1/h Mpc

Abstract: (abridged) We present measurements of galaxy-galaxy lensing from early commissioning imaging data from the Sloan Digital Sky Survey (SDSS). We measure a mean tangential shear around a stacked sample of foreground galaxies in three bandpasses out to angular radii of 600'', detecting the shear signal at very high statistical significance. The shear profile is well described by a power-law. A variety of rigorous tests demonstrate the reality of the gravitational lensing signal and confirm the uncertainty estimates. We interpret our results by modeling the mass distributions of the foreground galaxies as approximately isothermal spheres characterized by a velocity dispersion and a truncation radius. The velocity dispersion is constrained to be 150-190 km/s at 95% confidence (145-195 km/s including systematic uncertainties), consistent with previous determinations but with smaller error bars. Our detection of shear at large angular radii sets a 95% confidence lower limit $s>140^{\prime\prime}$, corresponding to a physical radius of $260h^{-1}$ kpc, implying that galaxy halos extend to very large radii. However, it is likely that this is being biased high by diffuse matter in the halos of groups and clusters. We also present a preliminary determination of the galaxy-mass correlation function finding a correlation length similar to the galaxy autocorrelation function and consistency with a low matter density universe with modest bias. The full SDSS will cover an area 44 times larger and provide spectroscopic redshifts for the foreground galaxies, making it possible to greatly improve the precision of these constraints, measure additional parameters such as halo shape, and measure the properties of dark matter halos separately for many different classes of galaxies.

Growing Spatial Correlations of Particle Displacements in a Simulated Liquid on Cooling toward the Glass Transition

Abstract: We define a correlation function that quantifies the spatial correlation of single-particle displacements in liquids and amorphous materials. We show that for an equilibrium liquid this function is related to fluctuations in a bulk dynamical variable. We evaluate this function using computer simulations of an equilibrium glass-forming liquid, and show that long range spatial correlations of displacements emerge and grow on cooling toward the mode coupling critical temperature. [S0031-9007(99)09452-1] Liquids cooled toward their glass transition exhibit remarkable dynamical behavior [1]. The initial slowing of transport processes for liquids at temperatures T well above their glass transition temperature Tg is described by the mode coupling theory (MCT) [2], which predicts diverging relaxation times at a dynamical critical temperature Tc (in real and simulated liquids, this divergence is only apparent). The dynamical singularity of MCT occurs without a corresponding growing static correlation length associated with density or composition fluctuations [3]. Yet recent studies show that in the range of T described by MCT, simulated glass-forming liquids exhibit spatially heterogeneous dynamics [4 ‐6]. In this Letter, we define a correlation function that quantifies the spatial correlation of particle displacements and evaluate this function for a Lennard-Jones liquid. We find that spatial correlations of displacement arise and become long ranged on cooling toward Tc. First, we briefly review the conventional static correla

Use of SAXS and linear correlation functions for the determination of the crystallinity and morphology of semi-crystalline polymers. Application to linear polyethylene

Abstract: The use of correlation functions to obtain the morphological parameters of crystalline-amorphous two-phase lamellar systems is critically reviewed and extended. It is shown that processing of the experimental SAXS-patterns only significantly affects the curvature of the autocorrelation triangle and that the parameters of the corresponding ideal two-phase structure can be determined independently of the data processing procedure. The methods to be used depend on the normalization of the correlation function. The validity of the formulation is illustrated for a sample of linear polyethylene, cooled and heated at 10°C per min. Crystallite thickening during crystallization and surface melting during heating are observed. The overall crystallinity and the fraction of semi-crystalline stacks during crystallization and melting are determined quantitatively as a function of temperature using the total scattering power of the corresponding ideal two-phase structure, correlation functions, and a scaling procedure. Absolute intensities are not required. The SAXS results are confirmed by independent techniques (DSC, WAXD, and SALLS). During crystallization, amorphous regions are present outside the semi-crystalline regions because growing spherulites do not fill space completely. During melting, larger amorphous regions develop in the spherulites because of the complete melting of stacks. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 1715–1738, 1999

Properties of galaxy clusters: mass and correlation functions

Abstract: We analyse parallel N-body simulations of three Cold Dark Matter (CDM) universes to study the abundance and clustering of galaxy clusters. The simulation boxes are 500h 1 Mpc on a side and cover a volume comparable to that of the forthcoming Sloan Digital Sky Survey. The use of a treecode algorithm and 47 million particles allows us at the same time to achieve high mass and force resolution. We are thus able to make robust measurements of cluster properties with good number statistics up to a redshift larger than unity. We extract halos using two independent, public domain group finders designed to identify virialised objects – ‘Friends-of-Friends’ (Davis et al. 1985) and ‘HOP’ (Eisenstein & Hut 1998) – and find consistent results. The correlation function of clusters as a function of mass in the simulations is in very good agreement with a simple analytic prescription based upon a Lagrangian biasing scheme developed by Mo & White (1996) and the Press-Schechter (PS) formalism for the mass function. The correlation length of clusters as a function of their number density, the R0–Dc relation, is in good agreement with the APM Cluster Survey in our open CDM model. The critical density CDM model (SCDM) shows much smaller correlation lengths than are observed. We also find that the correlation length does not grow as rapidly with cluster separation in any of the simulations as suggested by the analysis of very rich Abell clusters. Our SCDM simulation shows a robust deviation in the shape and evolution of the mass function when compared with that predicted by the PS formalism. Critical models with a low �8 normalization or small shape parameter have an excess of massive clusters compared with the PS prediction. When cluster normalized, the SCDM universe at z = 1 contains 10 times more clusters with temperatures greater than 7keV, compared with the Press & Schechter prediction. The agreement between the analytic and N-body mass functions can be improved, for clusters hotter than 3

Correlation functions for atmospheric data analysis

Abstract: Atmospheric data assimilation techniques rely on parametric models for spatial correlation functions. This article proposes and discusses various families of homogeneous and isotropic correlation models on Euclidean spaces and on the sphere. In particular, three simply parametrized classes of compactly supported, smooth, and analytically simple correlation functions are proposed. the first two classes approximate standard second- and third-order autoregressive functions, and a member of the third family approximates the Gaussian function within a maximal error of 0.0056. Furthermore, correlation models suggested previously for meteorological applications are checked for permissibility, with both positive and negative results.

Theory of Non-linear Susceptibility and Correlation Length in Glasses and Liquids

Abstract: Within the framework of the effective potential theory of the structural glass transition, we calculate for the p-spin model a static nonlinear susceptibility related to a four-point density correlation function, and show that it grows and diverges in mean field with exponent $\gamma=1/2$ as the mode coupling critical temperature T_c is approached from below. When T_c is approached from above, we calculate within the mode coupling framework a dynamic nonlinear susceptibility and show that there is a characteristic time where the susceptibility is a maximum, and that this time grows with decreasing T. We find that this susceptibility diverges as T_c is approached from above, and has key features in common with the ``displacement-displacement susceptibility'' recently introduced to measure correlated particle motion in simulations of glass-forming liquids.

Hyperextended Cosmological Perturbation Theory: Predicting Nonlinear Clustering Amplitudes

Abstract: We consider the long-standing problem of predicting the hierarchical clustering amplitudes Sp in the strongly nonlinear regime of gravitational evolution. N-body results for the nonlinear evolution of the bispectrum (the Fourier transform of the three-point density correlation function) suggest a physically motivated Ansatz that yields the strongly nonlinear behavior of the skewness, S3, starting from leading-order perturbation theory. When generalized to higher order (p > 3) polyspectra or correlation functions, this Ansatz leads to a good description of nonlinear amplitudes in the strongly nonlinear regime for both scale-free and cold dark matter models. Furthermore, these results allow us to provide a general fitting formula for the nonlinear evolution of the bispectrum that interpolates between the weakly and strongly nonlinear regimes, analogous to previous expressions for the power spectrum.

Correlation induced by Tsallis’ nonextensivity

Abstract: In Tsallis’ generalized statistical mechanics, correlation is induced by nonextensivity even if the microscopic degrees of freedom are dynamically independent. Here, using the classical ideal gas model, the generalized variance, covariance and correlation coefficient regarding the particle energies are calculated and their properties discussed. It is shown that the correlation is suppressed for a large number of particles. This demonstrates the validity of the independent particle picture for a dense gas rather than for a dilute gas. It is also found that, in the thermodynamic limit, the correlation again vanishes and the generalized variance exhibits a power-law behavior with respect to the particle number density. Relevance of these results to the zeroth law of thermodynamics in nonextensive statistical mechanics is pointed out.

Monitoring protein assembly using quasielastic light scattering spectroscopy.

Abstract: This article discussed the principles and practice of QLS with respect to protein assembly reactions. Particles undergoing Brownian motion in solution produce fluctuations in scattered light intensity. We have described how the temporal correlation function of these fluctuations can be measured and how mathematical analysis of the correlation function provides information about the distribution of diffusion coefficients of the particles. We have explained that deconvolution of the correlation function is an "ill-posed" problem and therefore that careful attention must be paid to the assumptions incorporated into data analysis procedures. We have shown how the Stokes-Einstein relationship can be used to convert distributions of diffusion coefficients into distributions of particle size. In the case of fibrillar polymers, this process allows direct determination of fibril length, enabling nucleation and elongation rates to be calculated. Finally, we have used examples from studies of A beta fibrillogenesis to illustrate the power these quantitative capabilities provide for understanding the molecular mechanisms of the fibrillogenesis reaction and for guiding the development of fibrillogenesis inhibitors.

The Spectral Correlation Function: A New Tool for Analyzing Spectral Line Maps

Abstract: The "spectral correlation function" analysis we introduce in this paper is a new tool for analyzing spectral line data cubes Our initial tests, carried out on a suite of observed and simulated data cubes, indicate that the spectral correlation function (SCF) is likely to be a more discriminating statistic than other statistical methods normally applied The SCF is a measure of similarity between neighboring spectra in the data cube When the SCF is used to compare a data cube consisting of spectral line observations of the interstellar medium (ISM) with a data cube derived from MHD simulations of molecular clouds, it can find differences that are not found by other analyses The initial results presented here suggest that the inclusion of self-gravity in numerical simulations is critical for reproducing the correlation behavior of spectra in star-forming molecular clouds

Accurate reaction rate calculations including internal and rotational motion: A statistical multi-configurational time-dependent Hartree approach

Abstract: A statistical approach to the direct calculation of thermal rate constants and cumulative reaction probabilities based on flux correlation functions is presented. It facilitates the accurate treatment of rotational and low frequency vibrational motion in rate constant calculations. Within this approach, a statistical sampling scheme is used to evaluate the trace in the flux correlation function and the multi-configurational time-dependent Hartree (MCTDH) approach is employed to simulate the system dynamics. The OH+Cl→O+HCl reaction is studied to demonstrate the efficiency of the approach. Known results for vanishing total angular, J=0, are reproduced. Moreover, thermal rate constants are calculated including all internal and rotational degrees of freedom explicitly, i.e., without invoking a J-shifting or K-conserving approximation. Based on these results, the accuracy of the J-shifting approximation is discussed.

"Wave transport in random media: The ballistic to diffusive transition"

Abstract: The character of wave transport through a strongly scattering medium, excited by a pulsed plane-wave source, is investigated as a function of sample thickness over the range from about one to 13 mean free paths. To examine the behavior theoretically, we perform a first-principles calculation of both the frequency correlation function of the transmitted field and the time-domain profile of the transmitted intensity. These quantities are investigated experimentally using an ultrasonic technique, which allows us to separate the ballistic and scattered components of the total transmitted field, and hence to measure the scattered component unambiguously in thin samples. For sample thicknesses greater than about four mean free paths, we find good agreement between our theory, the diffusion approximation, and our experimental data for both the frequency correlation function and the intensity time profile. In thinner samples, there are systematic differences between theory and experiment. To characterize the transition from ballistic to diffusive behavior in thin samples, we focus on the arrival time of the peak in the scattered component of the transmitted intensity; unexpectedly we find that the scattered peak arrival time exhibits an abrupt crossover between ballistic and diffusive behavior when the ratio of sample thickness to mean free path, L/l, is approximately equal to 3. Excellent agreement is obtained between our theory and experiment for this crossover behavior over the entire range of sample thicknesses investigated.

Reconstruction of random media using Monte Carlo methods.

Abstract: A simulated annealing algorithm is applied to the reconstruction of two-dimensional porous media with prescribed correlation functions. The experimental correlation function of an isotropic sample of Fontainebleau sandstone and a synthetic correlation function with damped oscillations are used in the reconstructions. To reduce the numerical effort we follow a proposal suggesting the evaluation of the correlation functions only along certain directions. The results show that this simplification yields significantly different microstructures as compared to a full evaluation of the correlation function. In particular, we find that the simplified reconstruction method introduces an artificial anisotropy that is originally not present.

The Spectral Correlation Function -- A New Tool for Analyzing Spectral-Line Maps

Abstract: The "spectral correlation function" analysis we introduce in this paper is a new tool for analyzing spectral-line data cubes. Our initial tests, carried out on a suite of observed and simulated data cubes, indicate that the spectral correlation function [SCF] is likely to be a more discriminating statistic than other statistical methods normally applied. The SCF is a measure of similarity between neighboring spectra in the data cube. When the SCF is used to compare a data cube consisting of spectral-line observations of the ISM with a data cube derived from MHD simulations of molecular clouds, it can find differences that are not found by other analyses. The initial results presented here suggest that the inclusion of self-gravity in numerical simulations is critical for reproducing the correlation behavior of spectra in star-forming molecular clouds.

On effects of resolution in dissipationless cosmological simulations

Abstract: We present a study of numerical effects in dissipationless cosmological simulations. The numerical effects are evaluated and studied by comparing results of a series of 64^3-particle simulations of varying force resolution and number of time steps, performed using three different N-body techniques: the Particle Mesh (PM), the adaptive P3M (AP3M) code, and the Adaptive Refinement Tree (ART) code. This study can therefore be interesting both as an analysis of numerical effects and as a systematic comparison of different codes. We find that the AP3M and the ART codes produce similar results, given that convergence is reached within the code type. We also find that numerical effects may affect the high-resolution simulations in ways that have not been discussed before. In particular, our study revealed the presence of two-body scattering, effects of which can be greatly amplified by inaccuracies of time integration. This process appears to affect the correlation function of matter, mass function and inner density of dark matter halos and other statistics at scales much larger than the force resolution, although different statistics may be affected in different fashion. We discuss the conditions at which strong two-body scattering is possible and discuss the choice of the force resolution and integration time step. Furthermore, we discuss recent claims that simulations with force softening smaller than the mean interparticle separation are not trustworthy and argue that this claim is incorrect in general and applies only to the phase-sensitive statistics. Our conclusion is that, depending on the choice of mass and force resolution and integration time step, a force resolution as small as 0.01 of the mean interparticle separation can be justified.

Direct observation of stretched-exponential relaxation in low-temperature lennard-jones systems using the cage correlation function

Abstract: We report on the direct observation of stretched exponential relaxation in low-temperature monatomic Lennard-Jones systems which were cooled slowly from the liquid phase to form crystals with a large number of defects. We use the cage correlation function [E. Rabani, J. D. Gezelter, and B. J. Berne, J. Chem. Phys. 107, 6867 (1997)] which measures changes in atomic surroundings to observe the stretched exponential relaxations. We obtain a distribution of hopping rates assuming that the origin of the Kohlrausch-Williams-Watts law is from static disorder in the distribution of barrier heights. [S0031-9007(99)09049-3]

Three-point velocity correlation functions in two-dimensional forced turbulence.

Abstract: We present a simple exact formula for three-point velocity correlation functions in two-dimensional turbulence which is valid at all scales and which interpolates between the direct and inverse cascade regimes As expected, these correlation functions are universal in these extreme regimes We also discuss the effects of anisotropy and friction

Cosmological constraints from the clustering properties of the X-ray Brightest Abell-type Cluster sample

Abstract: We present an analysis of the two-point correlation function, ξ (r), of the X-ray Brightest Abell-type Cluster sample (XBACs) of Ebeling et al. and of the cosmological constraints that it provides. If ξ (r) is modelled as a power-law, ξ (r)=(r0/r)γ, we find r0≃ 26.0 ± 4.5 h-1 Mpc and γ≃ 2.0 ± 0.4, with errors corresponding to 2 σ uncertainties for one significant fitting parameter. As a general feature, ξ (r) is found to remain positive up to r ≃ 50–55 h-1 Mpc, after which it declines and crosses zero. Only a marginal increase of the correlation amplitude is found as the flux limit is increased from 5 × 10-12 to 12 × 10-12 erg s-1 cm-2, thus indicating a weak dependence of the correlation amplitude on the cluster X-ray luminosity. Furthermore, we present a method to predict correlation functions for flux-limited X-ray cluster samples from cosmological models. The method is based on the analytical recipe by Mo & White and on an empirical approach to convert cluster fluxes into masses. We use a maximum likelihood method to place constraints on the model parameter space from the XBACs ξ (r). For scale-free primordial spectra, we find that the shape parameter of the power spectrum is determined to lie in the 2 σ range 0.05 ≤Γλ 0.20. As for the amplitude of the power spectrum, we find σ8≃ 0.4–0.8 for Ο0=1 and σ8≃ 0.8–2.0 for Ο0=0.3. The latter result is in complete agreement with, although less constraining than, results based on the local cluster abundance.

Local chain ordering in amorphous polymer melts: influence of chain stiffness

Abstract: Molecular dynamics simulation of a generic polymer model is applied to study melts of polymers with different types of intrinsic stiffness. Important static observables of the single chain such as gyration radius or persistence length are determined. Additionally we investigate the overall static melt structure including pair correlation function, structure function and orientational correlation function.

The correlation of peaks in the microwave background

Abstract: We present accurate small-angle predictions of the correlation function of hotspots in the microwave background radiation for Gaussian theories such as those predicted in most inflation models. The correlation function of peaks above a certain threshold depends only on the threshold and the power spectrum of temperature fluctuations. Since these are both potentially observable quantities in a microwave background map, there are no adjustable parameters in the predictions. These correlations should therefore provide a powerful test of the Gaussian hypothesis, and provide a useful discriminant between inflation and topological defect models such as the cosmic string model. The correlations have a number of oscillatory features, which should be detectable at high signal-to-noise ratio with future satellite experiments such as MAP and Planck.

Dynamics of pairwise motions

Abstract: We propose a simple closed-form expression relating v12(r)—the mean relative velocity of pairs of galaxies at fixed separation r—to the two-point correlation function of mass density fluctuations, ξ(r). Our Ansatz is an interpolation between the perturbative and stable clustering expressions for v12. We compare our analytic model for v12(r) with N-body simulations and find excellent agreement in the entire dynamical range probed by the simulations (0.1 ξ 1000). Our results can be used to estimate the cosmological density parameter, Ω, directly from redshift-distance surveys.

Removal of cooperativity in glass-forming materials to reveal the primitive -relaxation of the coupling model

Abstract: The microscopic short time dynamics of glass-forming materials seen by quasielastic neutron scattering, dynamic light scattering, high frequency dielectric relaxation and molecular dynamics simulation in the picosecond region show the existence of the primitive relaxation, exp(-t/τ 0 ), for t < t c , where t c 2 ps is the cross-over time of the coupling model. However, such an interpretation is not unique because the same data have also been considered to be evidence of the fast β-relaxation of the model coupling theory. An apparently unique test of the coupling model is via the macroscopic long time relaxation data of ortho-terphenyl confined in nanometre glass pores with a size that is smaller than the length-scale of cooperative motions of the molecules. Under this condition, motions of the majority of the molecules cease to be cooperative and the experimentally observed relaxation time is the primitive relaxation time, τ 0 , of the coupling model. On the other hand, τ 0 can also be obtained from relaxation data of bulk ortho-terphenyl fitted to a stretched exponential correlation function, exp[-(t/τ) 1-n ], by the relation τ = [t -n c τ 0 ] 1/(1-n) of the coupling model. Good agreement found between these two independently determined values of τ 0 provides strong support of the coupling model. A similar test of the coupling model is to compare the Johari-Goldstein β-relaxation time, τ β , with the calculated τ 0 at temperatures above the glass transition temperature. Both the β-relaxation and the primitive relaxation time being noncooperative in nature implies that τ β is comparable in order of magnitude to that of the calculated τ 0 , which is shown to be true for the fragile glass-former ortho-terphenyl as well as a non-fragile glass-former, cyclo-octanol.

The redshift evolution of clustering in the Hubble Deep Field

Abstract: We present a correlation function analysis for the catalogue of photometric redshifts obtained from the Hubble Deep Field image by Fernandez-Soto, Lanzetta & Yahil. By dividing the catalogue into redshift bins of width Δz=0.4 we measured the angular correlation function w(θ) as a function of redshift up to z∼4.8. From these measurements we derive the trend of the correlation length r0. We find that r0(z) is roughly constant with look-back time up to z≃2, and then increases to higher values at z≳2.4. We estimate the values of r0, assuming ξ(r,z)=[rr0(z)]−γ, γ=1.8 and various geometries. For Ω0=1 we find r0(z=3)≃7.00±4.87 h−1 Mpc, in good agreement with the values obtained from analysis of the Lyman break galaxies.