Showing papers on "Correlation function (statistical mechanics) published in 2001"

Template Matching using Fast Normalized Cross Correlation

Abstract: In this paper, we present an algorithm for fast calculation of the normalized cross correlation and its application to the problem of template matching. Given a template t, whose position is to be determined in an image f, the basic idea of the algorithm is to represent the template, for which the normalized cross correlation is calculated, as a sum of rectangular basis functions. Then the correlation is calculated for each basis function instead of the whole template. The result of the correlation of the template t and the image f is obtained as the weighted sum of the correlation functions of the basis functions. Depending on the approximation, the algorithm can by far outperform Fourier-transform based implementations of the normalized cross correlation algorithm and it is especially suited to problems, where many different templates are to be found in the same image f.

A comparison of various commonly used correlation functions for describing total scattering

Abstract: Total scattering, an increasingly important crystallographic research area, is defined theoretically in terms of correlation functions. Different researchers use different definitions for these functions, frequently leading to confusion in the literature. Here, a consistent set of equations for total-scattering correlation functions are developed and explicitly compared with other, often encountered, definitions. It is hoped that this will lead to increased transparency for newcomers to the field of total scattering.

Correlation modelling on the sphere using a generalized diffusion equation

Abstract: An important element of a data assimilation system is the statistical model used for representing the correlations of background error. This paper describes a practical algorithm that can be used to model a large class of two- and three-dimensional, univariate correlation functions on the sphere. Application of the algorithm involves a numerical integration of a generalized diffusion-type equation (GDE). The GDE is formed by replacing the Laplacian operator in the classical diffusion equation by a polynomial in the Laplacian. The integral solution of the GDE defines, after appropriate normalization, a correlation operator on the sphere. The kernel of the correlation operator is an isotropic correlation function. The free parameters controlling the shape and length-scale of the correlation function are the products k p T, p = 1, 2,..., where (-1) p κ p is a weighting ('diffusion') coefficient (Kp > 0) attached to the Laplacian with exponent p, and T is the total integration time'. For the classical diffusion equation (a special case of the GDE with κ p = 0 for all p > 1) the correlation function is shown to be well approximated by a Gaussian with length-scale equal to (2κ 1 T) 1/2 The Laplacian-based correlation model is particularly well suited for ocean models as it can be easily generalized to account for complex boundaries imposed by coastlines. Furthermore, a one-dimensional analogue of the GDE can be used to model a family of vertical correlation functions, which in combination with the two-dimensional GDE forms the basis of a three-dimensional, (generally) non-separable correlation model. Generalizations to account for anisotropic correlations are also possible by stretching and/or rotating the computational coordinates via a diffusion' tensor. Examples are presented from a variational assimilation system currently under development for the OPA ocean general-circulation model of the Laboratoire d'Oceanographie Dynamique et de Climatologie.

Weak transition matrix elements from finite volume correlation functions

Abstract: The two-body decay rate of a weakly decaying particle (such as the kaon) is shown to be proportional to the square of a well-defined transition matrix element in finite volume. Contrary to the physical amplitude, the latter can be extracted from finite-volume correlation functions in euclidean space without analytic continuation. The K→ππ transitions and other non-leptonic decays thus become accessible to established numerical techniques in lattice QCD.

Bayesian estimation of semi‐parametric non‐stationary spatial covariance structures

Abstract: We use the Sampson and Guttorp approach to model the non-stationary correlation function r(x, x′) of a Gaussian spatial process through a bijective space deformation, f, so that in the deformed space the spatial correlation function can be considered isotropic, namely r(x, x′) = ρ(∣ f(x)−f(x′)∣), where ρ belongs to a known parametric family. Given the locations in the deformed space of a number of geographic sites at which data are available, we smoothly extrapolate the deformation to the whole region of interest. Using a Bayesian framework, we estimate jointly these locations, as well as the parameters of the correlation function and the variance parameters. The advantage of our Bayesian approach is that it allows us to obtain measures of uncertainty of all these parameters. As the parameter space is of a very high dimension, we implement an MCMC method for obtaining samples from the posterior distributions of interest. We demonstrate our method through a simulation study, and show an application to a real data set. Copyright © 2001 John Wiley & Sons, Ltd.

Linear and non-linear contributions to pairwise peculiar velocities

Abstract: We write the correlation function of dark matter particles, ξ(r), as the sum of two terms – one which accounts for non-linear evolution, and dominates on small scales, and another which is essentially the term from linear theory, and dominates on large scales. We use models of the number and spatial distribution of haloes and halo density profiles to describe the non-linear term and its evolution. The result provides a good description of the evolution of ξ(r) in simulations. We then use this decomposition to provide simple and accurate models of how the single-particle velocity dispersion evolves with time, and how the first and second moments of the pairwise velocity distribution depend on scale. The key idea is to use the simple physics of linear theory on large scales, the simple physics of the virial theorem on small scales and our model for the correlation function to tell us how to weight the two types of contributions (linear and non-linear) to the pairwise velocity statistics. When incorporated into the streaming model, our results will allow a simple accurate description of redshift-space distortions over the entire range of linear to highly non-linear regimes.

Debye-Waller Factor of Liquid Silica: Theory and Simulation

Abstract: We show that the prediction of mode-coupling theory for a model of a network-forming strong glass former correctly describes the wave-vector dependence of the Debye-Waller factor. To obtain a good description it is important to take into account the triplet correlation function c(3), which we evaluate from a computer simulation. Our results support the possibility that this theory is able to describe accurately the nonergodicity parameters of simple as well as of network-forming liquids.

Euler-Poincaré characteristics of classes of disordered media

Abstract: We consider a family of statistical measures based on the Euler-Poincar\'e characteristic of n-dimensional space that are sensitive to the morphology of disordered structures. These measures embody information from every order of the correlation function but can be calculated simply by summing over local contributions. We compute the evolution of the measures with density for a range of disordered microstructural models; particle-based models, amorphous microstructures, and cellular and foamlike structures. Analytic results for the particle-based models are given and the computational algorithm verified. Computational results for the different microstructures exhibit a range of qualitative behavior. A length scale is derived based on two-point autocorrelation functions to allow qualitative comparison between the different structures. We compute the morphological parameters for the experimental microstructure of a sandstone sample and compare them to three common stochastic model systems for porous media. None of the statistical models are able to accurately reproduce the morphology of the sandstone.

Extraction of domain structure information from small-angle scattering patterns of bulk materials

Abstract: A method is presented that permits the extraction and visualization of topological domain structure information contained in small-angle scattering (SAS) patterns without complex pretreatment. Multi-dimensional noisy raw data can be processed. Such data are, for instance, accumulated in the field of materials research from short-exposure-time in situ small-angle X-ray scattering (SAXS) experiments with synchrotron radiation. The result is a multi-dimensional intersect or chord distribution, which is defined as the Laplacian of the correlation function. Moreover, it is equivalent to the autocorrelation of the gradient of the electron density. The procedure is, in particular, adapted to the analysis of the nanoscale structure of samples with fibre symmetry, such as polymer fibres or strained elastomers. Multi-dimensional relations among morphological components become apparent in real space and help to elucidate the nature of the processes governing formation and change of structure on the nanometre scale. Utilizing digital signal processing tools, the algorithm is based on spatial frequency filtering of the raw data. The background to be subtracted from the small-angle scattering pattern is formed from its own low spatial frequencies. Noise may be removed by suppressing high spatial frequencies. In the frequency band between these low and high spatial frequencies, the domain structure information of the studied nanocomposite appears.

Determining the solvation correlation function from three-pulse photon echoes in liquids

Abstract: The decay of three-pulse photon echo signals from a solute in a liquid solvent is sensitive to the solute’s transition frequency fluctuations, as characterized by its two-point time correlation function, otherwise known as the solvation correlation function. The most widely used method for determining this solvation correlation function from photon echo data involves the three-pulse photon echo peak shift (3PEPS) method. Using this method the long-time decay of the solvation correlation function can be obtained directly, but the determination of the short-time decay requires a difficult numerical fitting procedure. In this study we propose several alternative approaches to determining the solvation correlation function from echo data, the most promising and straightforward of which we call the S3PE (short-time slope of the three-pulse photon echo) method. The accuracy and efficacy of this approach is illustrated by extracting the solvation correlation function from “experimental” data obtained from classi...

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Abstract: Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.

Inhomogeneity and anisotropy effects on the redistribution term in Reynolds-averaged Navier–Stokes modelling

Abstract: A channel flow DNS database at Re τ = 590 is used to assess the validity of modelling the redistribution term in the Reynolds stress transport equations by elliptic relaxation. The model assumptions are found to be globally consistent with the data. However, the correlation function between the fluctuating velocity and the Laplacian of the pressure gradient, which enters the integral equation of the redistribution term, is shown to be anisotropic. It is elongated in the streamwise direction and strongly asymmetric in the direction normal to the wall, in contrast to the isotropic, exponential model representation used in the original elliptic relaxation model. This discrepancy is the main cause of the slight amplification of the energy redistribution in the log layer as predicted by the elliptic relaxation equation. New formulations of the model are proposed in order to correct this spurious behaviour, by accounting for the rapid variations of the length scale and the asymmetrical shape of the correlation function. These formulations do not rely on the use of so-called ‘wall echo’ correction terms to damp the redistribution. The belief that the damping is due to the wall echo effect is called into question through the present DNS analysis.

Correlation functions of the higher spin XXX chains

Abstract: Using the algebraic Bethe ansatz, we consider the correlation functions of the integrable higher spin chains. We apply a method recently developed for the spin ½ Heisenberg chain, based on the solution of the quantum inverse problem. We construct a representation for the correlation functions on a finite chain for arbitrary spin. Then we show how the string solutions of the Bethe equations can be considered in the framework of this approach in the thermodynamic limit. Finally, a multiple integral representation for the spin 1 zero-temperature correlation functions is obtained in the thermodynamic limit.

Hopping in the glass configuration space: Subaging and generalized scaling laws

Abstract: Aging dynamics in glassy systems is investigated by considering the hopping motion in a rugged energy landscape whose deep minima are characterized by an exponential density of states. In particular we explore the behavior of a generic two-time correlation function Pi(t_w+t,t_w) below the glass transition temperature T_g when both the observation time t and the waiting time t_w become large. We show the occurrence of ordinary scaling behavior, which includes normal aging and subaging, and the possible simultaneous occurrence of generalized scaling behavior. Which situation occurs depends on the form of the effective transition rates between the low lying states. Employing a ``Partial Equilibrium Concept'', the exponents of the aging and the asymptotic form of the scaling functions are obtained both by simple scaling arguments and by analytical calculations. The predicted scaling properties compare well with Monte-Carlo simulations in dimensions d = 1-1000 and it is argued that a mean-field type treatment of the hopping motion fails to describe the aging dynamics in any dimension. Implications for more general situations involving different forms of transition rates and the occurrence of many scaling regimes in the t-t_w plane are pointed out.

Near-field intensity correlations of scattered light

Abstract: We show that the two-point correlation function in the near field of scattered light is simply related to the scattered intensity distribution. We present a new, to our knowledge, optical scheme to measure the correlation function in the near field, and we describe a processing technique that permits the subtraction of stray light on a statistical basis. We present experimental data for solutions of latex spheres, and we show that this novel technique is a powerful alternative to static light scattering.

Incoherent bistatic scattering from the sea surface at L-band

Abstract: A bistatic electromagnetic wave scattering model for the sea surface is developed to examine its wind dependence property over a wide range of incident angles along the specular direction. This is done by combining an existing scattering model with a sea spectrum recently reported in the literature. In general, electromagnetic wave scattering from a rough surface is dependent on the Fourier transform of the nth power of its height correlation function which can be computed numerically from the surface spectrum. This transform relation indicates that scattering is sensitive not only to the surface spectrum but also to its convoluted properties. Generally, surface scattering is sensitive only to a portion of the surface correlation measured from the origin. The size of this portion is a function of three variables (the incident angle, the surface height standard deviation, and the exploring wavelength) and the rate of decay of the correlation function. The decay rate near the origin of the sea surface correlation is very small, so much so that at L-band this portion is too wide for a two-term approximation of the correlation function. This is true in spite of the fact that the sea surface has a very large rms height. Thus, a scattering model based on geometric optics is generally not applicable at L-band especially at large angles of incidence. An additional finding is that in specular scattering wind dependence is stronger at larger angles of incidence for incident angles between 0 and 70/spl deg/ over the wind speed range of 4 m/s-20 m/s.

Elastic constants from direct correlation functions in nematic liquid crystals: A computer simulation study

Abstract: Density functional theories such as the Poniewierski–Stecki theory relate the elastic properties of nematic liquid crystals with their local liquid structure, i.e., with the direct correlation function (DCF) of the particles. We propose a way to determine the DCF in the nematic state from simulations without any approximations, taking into account the dependence of pair correlations on the orientation of the director explicitly. Using this scheme, we evaluate the Frank elastic constants K11, K22, and K33 in a system of soft ellipsoids. The values are in good agreement with those obtained directly from an analysis of order fluctuations. Our method thus establishes a reliable way to calculate elastic constants from pair distributions in computer simulations.

Continuum in the spin-excitation spectrum of a haldane chain observed by neutron scattering in CsNiCl3.

Abstract: The spin-excitation continuum, expected to dominate the low-energy fluctuation spectrum in the Haldane spin chain around the Brillouin zone center, q = 0, is directly observed by inelastic magnetic neutron scattering in the S = 1 quasi-1D antiferromagnet CsNiCl3. We find that the single mode approximation fails, and that a finite energy width appears in the dynamic correlation function S(q,omega) for q less, similar 0.5 pi. The width increases with decreasing q, while S(q,omega) acquires an asymmetric shape qualitatively similar to that predicted for the two-magnon continuum in the nonlinear sigma-model.

Semi‐analytic approach to understanding the distribution of neutral hydrogen in the Universe

Abstract: Analytic derivations of the correlation function and the column density distribution for neutral hydrogen in the intergalactic medium (IGM) are presented, assuming that the non-linear baryonic mass density distribution in the IGM is lognormal. This ansatz was used earlier by Bi & Davidsen to perform one-dimensional simulations of lines of sight and analyse the properties of absorption systems. We have taken a completely analytic approach, which allows us to explore a wide region of the parameter space for our model. The analytic results have been compared with observations to constrain various cosmological and IGM parameters, whenever possible. Two kinds of correlation functions are defined: (i) along the line of sight (LOS); and (ii) across the transverse direction. We find that the effects on the LOS correlation owing to changes in cosmology and the slope of the equation of state of the IGM, γ, are of the same order, which means that we cannot constrain both the parameters simultaneously. However, it is possible to constrain γ and its evolution using the observed LOS correlation function at different epochs provided that one knows the background cosmology. We suggest that the constraints on the evolution of γ obtained using the LOS correlation can be used as an independent tool to probe the reionization history of the Universe. From the transverse correlation function, we obtain the excess probability, over random, of finding two neutral hydrogen overdense regions separated by an angle θ. We find that this excess probability is always less than 1 per cent for redshifts greater than 2. Our models also reproduce the observed column density distribution for neutral hydrogen, and the shape of the distribution depends on γ. Our calculations suggest that one can rule out γ>1.6 for z≃2.31 using the column density distribution. However, one cannot rule out higher values of γ at higher redshifts.

Spin correlations in the two-leg antiferromagnetic ladder in a magnetic field

Abstract: We study the ground-state spin correlations in the gapless incommensurate regime of a $S=1/2$ $\mathrm{XXZ}$ chain and a two-leg antiferromagnetic ladder under a magnetic field, in which the gapless excitations form a Tomonaga-Luttinger (TL) liquid. We calculate numerically the two-spin correlation functions and the local magnetization in the two models using the density-matrix renormalization-group method. By fitting the numerical results for an open $\mathrm{XXZ}$ chain of 100 spins to correlation functions of a Gaussian model, we determine the TL-liquid parameter K and the amplitudes of the correlation functions. The value of K estimated from the fits is in excellent agreement with the exact value obtained from the Bethe ansatz. We apply the same method to the open ladder consisting of 200 spins and determine the dependence of K on the magnetization M. The $K\ensuremath{-}M$ relation changes drastically depending on the ratio of the coupling constants in the leg and rung directions. We also discuss implications of these results to experiments on the nuclear spin relaxation rate ${1/T}_{1}$ and dynamical spin structure factors.

Phase ordering in nematic liquid crystals.

Abstract: We study the kinetics of the nematic-isotropic transition in a two-dimensional liquid crystal by using a lattice Boltzmann scheme that couples the tensor order parameter and the flow consistently. Unlike in previous studies, we find that the time dependences of the correlation function, energy density, and number of topological defects obey dynamic scaling laws with growth exponents that, within the numerical uncertainties, agree with the value $1/2$ expected from simple dimensional analysis. We find that these values are not altered by the hydrodynamic flow. In addition, by examining shallow quenches, we find that the presence of orientational disorder can inhibit amplitude ordering.

The Effects of Noise and Sampling on the Spectral Correlation Function

Abstract: The effects of noise and sampling on the spectral correlation function (SCF) introduced by Rosolowsky and coworkers are studied using observational data, numerical simulations of magnetohydrodynamic turbulence, and simple models of Gaussian spectral line profiles. The most significant innovations of this paper are (1) the normalization of the SCF based on an analytic model for the effect of noise and (2) the computation of the SCF as a function of the spatial lag between spectra within a map. A new definition of the "quality" of a spectrum, Q, is introduced, which is correlated with the usual definition of signal-to-noise ratio. The prenormalization value of the SCF is a function of Q. We derive analytically the effect of noise on the SCF, and then normalize the SCF to its analytic approximation. By computing the dependence of the SCF on the spatial lag, S0(Δr), we have been able to conclude the following: (1) S0(Δr) is a power law, with slope α, in the range of scales li < l < lo. (2) The correlation outer scale, lo, is determined by the size of the map, and no evidence for a true departure from self-similarity on large scales has been found. (3) The correlation inner scale, li, is a true estimate of the smallest self-similar scale in a map. (4) The spectral slope, α, in a given region, is independent of velocity resolution (above a minimum resolution threshold), spatial resolution, and average spectrum quality. (5) Molecular transitions that trace higher gas density yield larger values of α (steeper slopes) than transitions tracing lower gas density. (6) Nyquist sampling, bad pixels in detector arrays, and reference-sharing data acquisition need to be taken into account for a correct determination of the SCF at Δr = 1. The value of α, however, can be computed correctly without a detailed knowledge of observational procedures.

Spin and current correlation functions in the d-density-wave state of the cuprates

Abstract: We calculate the spin-spin and current-current correlation functions in states exhibiting d_{{x^2}-{y^2}}-density wave (DDW) order, d_{{x^2}-{y^2}} superconducting order (DSC), or both types of order. The spin-spin correlation functions in a state with both DDW and DSC order and in a state with DDW order alone, respectively, illuminate the resonant peak seen in the superconducting state of the underdoped cuprates and the corresponding feature seen in the pseudogap regime. The current-current correlation function in a state with both DDW and DSC order evinces a superfluid density with doping dependence which is consistent with that of the underdoped cuprates. These calculations strengthen the identification of the pseudogap with DDW order and of the underdoped cuprates with a state with both DDW and DSC order.

Two-Point Descriptions of Wake Turbulence with Application to Noise Prediction

Abstract: Measurements of the four-dimensional two-point correlation tensor of a fully developed airfoil wake are presented. These data allow examination of the characteristic eddies of the turbulence through proper orthogonal decomposition and linear stochastic estimation. A simple generic technique has been developed to extrapolate the correlation tensor function from the Reynolds stress field based on the hypothesis that its form is largely determined by the constraints imposed by inhomogeneity and continuity. Estimates for the plane wake compare favorably with measurements, and, interestingly, proper orthogonal modes and characteristic eddy structures inferred from the estimates are similar to those obtained from measurements. This implies that these measures of the correlation function are largely determined by fairly basic physical information, suggesting some simplification of the turbulence-modeling problem for applications such as aeroacoustics.

On the bayesian approach to calculating time correlation functions in quantum systems; reaction dynamics and spectroscopy

Abstract: We assess the current status, advantages and limitations of the numerical analytic continuation approach to computing time correlation functions in large many-body quantum systems characteristic of condensed phase chemical processes. We determine the quantum correlation function as a function of complex time, and use its analytic properties to select a suitable contour in the complex time plane along which the function can be evaluated efficiently by stochastic simulation methods. The simulation data are then used to obtain the values of the correlation function along the real-time axis through a maximum entropy numerical analytic continuation procedure. This approach is used to compute the dynamical properties of several condensed phase processes including vibrational relaxation lineshapes and canonical reaction rates. We discuss how to improve the accuracy of the numerical analytic continuation methods.

Spectral diffusion in liquids with fluctuating solvent responses: Dynamical heterogeneity and rate exchange

Abstract: A recent theory for the time dependence of inhomogeneous line shapes is extended to account for fluctuations of the solvent response times τ in both space (heterogeneity) and time (rate exchange). Different simulation techniques are outlined for solving the Ornstein–Uhlenbeck type spectral diffusion in the situation expected for supercooled liquids. For intrinsically exponential solvent dynamics, slow rate exchange (or static heterogeneity) has to be assumed in order to reproduce measurements of the Stokes-shift correlation function C(t) and inhomogeneous linewidth σ(t) simultaneously.

Density functional theory of inhomogeneous fluid mixture based on bridge function

Abstract: A simple density functional theory is proposed for an inhomogeneous fluid mixture by approximating its one-particle correlation function in terms of the second-order direct correlation functions and the bridge function of the corresponding homogeneous system. The theory is applied to predict the structure of a binary hard sphere mixture as well as Lennard-Jones fluid mixture near a hard wall, and the calculated density profiles for both the components are shown to agree quite well with the corresponding computer simulation results for both the systems. This theory for an inhomogeneous fluid mixture is further applied to homogeneous hard sphere mixture as well as Lennard-Jones fluid mixture and the calculated radial distribution functions are found to compare quite well with the same obtained through integral equation theory of fluid mixture.

Coulomb Luttinger liquid

Abstract: Accurate expressions, valid in experimentally relevant regimes, are presented for the effect of a long-ranged Coulomb interaction on the low-energy properties (momentum distribution function, density of states, electron spectral function, and ${4k}_{F}$ correlation function) of one-dimensional electron systems. The importance of plasmon dispersion (as opposed to exponent) effects in the spectral function is demonstrated.

Probing exotic spin correlations by muon spin depolarization measurements with applications to spin glass dynamics

Abstract: We develop a method to probe the local spin dynamic autocorrelation function, using magnetic-field-dependent muon depolarization measurements. We apply this method to muon spin relaxation experiments in the dilute Heisenberg spin glass $\mathrm{Ag}\mathrm{Mn}$ $(p$ at. %) at $Tg{T}_{g},$ where the correlations of the Mn local magnetic moment are strongly nonexponential. Our results clearly indicate that the dynamics of this spin glass cannot be described by a distribution of correlation times. Therefore, we analyze the data assuming a local spin correlation function which is the product of a power law times a cutoff function. The concentration and temperature dependence of the parameters of this function are determined. Our major conclusion is that in the temperature region close to ${T}_{g}$ the correlation function is dominated by an algebraic relaxation term.