Showing papers on "Correlation function (statistical mechanics) published in 2007"

Quantum quenches in extended systems

Abstract: We study in general the time evolution of correlation functions in a extended quantum system after the quench of a parameter in the Hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d+1 dimensions. For d = 1 this allows us to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the Gaussian (mean field) approximation. These predictions are checked against the real time evolution of some solvable models that allow us also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments.

One-particle dynamical correlations in the one-dimensional Bose gas

Abstract: The momentum- and frequency-dependent one-body correlation function of the one-dimensional interacting Bose gas (Lieb–Liniger model) in the repulsive regime is studied using the Algebraic Bethe Ansatz and numerics. We first provide a determinant representation for the field form factor which is well adapted to numerical evaluation. The correlation function is then reconstructed to high accuracy for systems with finite but large numbers of particles, for a wide range of values of the interaction parameter. Our results are extensively discussed, in particular their specialization to the static case.

On the Calculation of Time Correlation Functions

Abstract: I11 . Time Correlation Functions and Memory Functions . . . . . . . 94 A . Projection Operators and the Memory Functions . . . . . . . 94 B . Memory Function Equation for Multivariate Processes . . . . . 100 C . The Modified Langevin Equation . . . . . . . . . . . 102 D . Continued Fraction Representation of Time-Correlation Functions . . 106 E . Dispersion Relations and Sum Rules for the Memory Function . . . 108

Flow‐adaptive moderation of spurious ensemble correlations and its use in ensemble‐based data assimilation

Abstract: Many ensemble Kalman filter (EnKF) data assimilation (DA) schemes reduce the effect of spurious ensemble correlations caused by small ensemble size by multiplying the correlations by moderation functions Moderation functions envelop true error correlation functions Ideal moderation functions would adapt to variations in the movement and width of true error correlation functions Here, we describe a new method in which flow-dependent moderation functions are built from powers of smoothed ensemble correlations The approach imparts to the moderation function movement and width information retained by the smoothed ensemble correlations Spurious smoothed ensemble correlations are attenuated by raising them to a power Simple systems were used to compare DA performance using such Smoothed ENsemble COrrelations Raised to a Power (SENCORP) moderation functions against DA performance using optimally tuned but non-adaptive moderation functions The simple systems considered feature propagating error correlation functions and error correlation functions with variable width It was found that when significant spatio-temporal variations in the true error correlation function are present, SENCORP moderation functions are superior to non-adaptive moderation functions In the absence of spatio-temporal variations, the DA performance of SENCORP moderation functions was found to be statistically indistinguishable from the DA performance of non-adaptive moderation functions An example using a primitive equation global model is given to illustrate how the method could be used to improve the performance of a local ensemble Kalman filter/smoother, particularly when larger observation volumes are used, to better account for error propagation and/or observations that represent vertically averaged variables Published in 2007 by John Wiley & Sons, Ltd

Ghost imaging with thermal light by third-order correlation

Abstract: Ghost imaging with classical incoherent light by third-order correlation is investigated. We discuss the similarities and the differences between ghost imaging by third-order correlation and by second-order correlation, and analyze the effect from each correlation part of the third-order correlation function on the imaging process. It is shown that the third-order correlated imaging includes richer correlated imaging effects than the second-order correlated one, while the imaging information originates mainly from the correlation of the intensity fluctuations between the test detector and each reference detector, as does ghost imaging by second-order correlation.

Observing the formation of long-range order during Bose-Einstein condensation.

Abstract: We have experimentally investigated the formation of off-diagonal long-range order in a gas of ultracold atoms. A magnetically trapped atomic cloud prepared in a highly nonequilibrium state thermalizes and thereby crosses the Bose-Einstein condensation phase transition. The evolution of phase coherence between different regions of the sample is constantly monitored and information on the spatial first-order correlation function is obtained. We observe the growth of the spatial coherence and the formation of long-range order in real time and compare it to the growth of the atomic density. Moreover, we study the evolution of the momentum distribution during the nonequilibrium formation of the condensate.

Anisotropic magnification distortion of the 3D galaxy correlation. I. Real space

Abstract: It has long been known that gravitational lensing, primarily via magnification bias, modifies the observed galaxy (or quasar) clustering. Such discussions have largely focused on the 2D angular correlation function. Here and in paper II [L. Hui, E. Gaztanaga, and M. LoVerde, arXiv:0710.4191] we explore how magnification bias distorts the 3D correlation function and power spectrum, as first considered by Matsubara [Astrophys. J. Lett. 537, L77 (2000).]. The interesting point is that the distortion is anisotropic. Magnification bias in general preferentially enhances the observed correlation in the line-of-sight (LOS) orientation, especially on large scales. For instance, at a LOS separation of {approx}100 Mpc/h, where the intrinsic galaxy-galaxy correlation is rather weak, the observed correlation can be enhanced by lensing by a factor of a few, even at a modest redshift of z{approx}0.35. This effect presents an interesting opportunity as well as a challenge. The opportunity: this lensing anisotropy is distinctive, making it possible to separately measure the galaxy-galaxy, galaxy-magnification, and magnification-magnification correlations, without measuring galaxy shapes. The anisotropy is distinguishable from the well-known distortion due to peculiar motions, as will be discussed in paper II. The challenge: the magnification distortion of the galaxy correlation must be accounted for in interpretingmore » data as precision improves. For instance, the {approx}100 Mpc/h baryon acoustic oscillation scale in the correlation function is shifted by up to {approx}3% in the LOS orientation, and up to {approx}0.6% in the monopole, depending on the galaxy bias, redshift, and number count slope. The corresponding shifts in the inferred Hubble parameter and angular diameter distance, if ignored, could significantly bias measurements of the dark energy equation of state. Lastly, magnification distortion offers a plausible explanation for the well-known excess correlations seen in pencil beam surveys.« less

Cross-correlation lensing: Determining galaxy and cluster mass profiles from statistical weak lensing measurements

Abstract: We present a new nonparametric method for determining mean 3D density and mass profiles from weak-lensing measurements around stacked samples of galaxies or clusters, that is, from measurement of the galaxy-shear or cluster-shear correlation functions Since the correlation function is statistically isotropic, this method evades problems, such as projection of large-scale structure along the line of sight or halo asphericity, that complicate attempts to infer masses from weak-lensing measurements of individual objects We demonstrate the utility of this method in measuring halo profiles, galaxy-mass and cluster-mass cross-correlation functions, and cluster virial masses We test this method on an N-body simulation and show that it correctly and accurately recovers the 3D density and mass profiles of halos We find no evidence of problems due to a mass sheet degeneracy in the simulation Cross-correlation lensing provides a powerful method for calibrating the mass-observable relation for use in measurement of the cluster mass function in large surveys It can also be used on large scales to measure and remove the halo bias and thereby provide a direct measurement of Ωmσ8

Investigation of q-dependent dynamical heterogeneity in a colloidal gel by x-ray photon correlation spectroscopy.

Abstract: We use time-resolved X-Photon Correlation Spectroscopy to investigate the slow dynamics of colloidal gels made of moderately attractive carbon black particles. We show that the slow dynamics is temporally heterogeneous and quantify its fluctuations by measuring the variance $\chi$ of the instantaneous intensity correlation function. The amplitude of dynamical fluctuations has a non-monotonic dependence on scattering vector $q$, in stark contrast with recent experiments on strongly attractive colloidal gels [Duri and Cipelletti, \textit{Europhys. Lett.} \textbf{76}, 972 (2006)]. We propose a simple scaling argument for the $q$-dependence of fluctuations in glassy systems that rationalizes these findings.

Electron correlation: the many-body problem at the heart of chemistry.

Abstract: The physical interactions among electrons and nuclei, responsible for the chemistry of atoms and molecules, is well described by quantum mechanics and chemistry is therefore fully described by the solutions of the Schrodinger equation. In all but the simplest systems we must be content with approximate solutions, the principal difficulty being the treatment of the correlation between the motions of the many electrons, arising from their mutual repulsion. This article aims to provide a clear understanding of the physical concept of electron correlation and the modern methods used for its approximation. Using helium as a simple case study and beginning with an uncorrelated orbital picture of electronic motion, we first introduce Fermi correlation, arising from the symmetry requirements of the exact wave function, and then consider the Coulomb correlation arising from the mutual Coulomb repulsion between the electrons. Finally, we briefly discuss the general treatment of electron correlation in modern electronic-structure theory, focussing on the Hartree-Fock and coupled-cluster methods and addressing static and dynamical Coulomb correlation.

Kinetic Theory of Coupled Oscillators

Abstract: We present an approach for the description of fluctuations that are due to finite system size induced correlations in the Kuramoto model of coupled oscillators. We construct a hierarchy for the moments of the density of oscillators that is analogous to the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy in the kinetic theory of plasmas and gases. To calculate the lowest order system size effect, we truncate this hierarchy at second order and solve the resulting closed equations for the two-oscillator correlation function around the incoherent state. We use this correlation function to compute the fluctuations of the order parameter, including the effect of transients, and compare this computation with numerical simulations.

Patchy sticky hard spheres: Analytical study and Monte Carlo simulations

Abstract: potential otherwise. We analyze the location of the fluid-fluid transition and of the percolation line as a function of the size of the patch the fractional coverage of the sphere’s surface and of the number of patches within a virial expansion up to third order and within the first two terms C0 and C1 of a class of closures Cn hinging on a density expansion of the direct correlation function. We find that the locations of the two lines depend sensitively on both the total adhesive coverage and its distribution. The treatment is almost fully analytical within the chosen approximate theory. We test our findings by means of specialized Monte Carlo simulations and find the main qualitative features of the critical behavior to be well captured in spite of the low density perturbative nature of the closure. The introduction of anisotropic attractions into a model suspension of spherical particles is a first step toward a more realistic description of globular proteins in solution. © 2007 American Institute of Physics. DOI: 10.1063/1.2805066

Time-resolved noise of adiabatic quantum pumps

Abstract: We investigate quantum-statistical correlation properties of a periodically driven mesoscopic scatterer on a time scale shorter than the period of a drive. In this limit the intrinsic quantum fluctuations in the system of fermions are the main source of a noise. Nevertheless the effect of a slow periodic drive is clearly visible in a two-time current-current correlation function as a specific periodic in time modulation. In the limit of a strong drive such a modulation can change the sign of a current correlation function.

Dimensional dependence of the metal-insulator transition

Abstract: We study the dependence on the spatial dimensionality of different quantities relevant in the description of the Anderson transition by combining numerical calculations in a $3\ensuremath{\leqslant}d\ensuremath{\leqslant}6$ disordered tight-binding model with theoretical arguments. Our results indicate that, in agreement with the one-parameter scaling theory, the upper critical dimension for localization is infinity. Typical properties of the spectral correlations at the Anderson transition such as level repulsion or a linear number variance are still present in higher dimensions though eigenvalue correlations get weaker as the dimensionality of the space increases. It is argued that such a critical behavior can be traced back to the exponential decay of the two-level correlation function in a certain range of eigenvalue separations. We also discuss to what extent different effective random matrix models proposed in the literature to describe the Anderson transition provide an accurate picture of this phenomenon.

Modeling Effects of Random Rough Interface on Power Absorption Between Dielectric and Conductive Medium in 3-D Problem

Abstract: We study the effects of a random rough surface on the power absorption between a dielectric and conductive medium in a 3-D configuration where the surface height varies in both horizontal directions. The analytic small perturbation method of second order and numerical T-matrix method are used. The absorption depends on the root mean square height, correlation length, and correlation function of the random rough surface. A closed-form expression of power absorption enhancement factor is obtained from small perturbation method of second order. Results show that the T-matrix method agrees with the small perturbation method for rough surfaces with a small slope. We further compare the 3-D results to the previous 2-D results and show significant difference. The power absorption enhancement factor exhibits saturation for the Gaussian correlation function, but not for the exponential correlation function

Characterizing wireless networks by spatial correlations

Abstract: We show that n-point correlation functions are powerful tools to characterize the structure of wireless networks. We argue that they convey the most interesting correlation structures in node locations, while being significantly easier and faster to compute than the classical correlation functions of stochastic geometry. As a case study we analyze the spatial structure of the WLAN access point locations in the east and west coasts of the USA. It is found that measured access point locations feature power-law or scale-free behavior in their correlation structures. Node location models commonly used in simulations and analytical calculations are also studied, and found to be unrealistic in this regard

Self-Similar Collisionless Shocks

Abstract: Observations of γ-ray burst afterglows suggest that the correlation length of magnetic field fluctuations downstream of relativistic nonmagnetized collisionless shocks grows with distance from the shock to scales much larger than the plasma skin depth. We argue that this indicates that the plasma properties are described by a self-similar solution and derive constraints on the scaling properties of the solution. For example, we find that the scaling of the characteristic magnetic field amplitude with distance from the shock is B DsB, with -1 < sB ≤ 0; that the spectrum of accelerated particles is dn/dE E−2/(sB + 1); and that the scaling of the magnetic correlation function is Bi(x)Bj(x + Δx) Δx2sB (for Δx D). We show that the plasma may be approximated as a combination of two self-similar components: a kinetic component of energetic particles and an MHD-like component representing thermal particles. We argue that the latter may be considered as infinitely conducting, in which case sB = 0 and the scalings are completely determined (e.g., dn/dE E-2 and B D0, with possible logarithmic corrections). Similar claims apply to nonrelativistic shocks such as in supernova remnants, if the upstream magnetic field can be neglected. Self-similarity has important implications for any model of particle acceleration and/or field generation. For example, we show that the diffusion function in the angle μ of momentum p in diffusive shock acceleration models must satisfy Dμμ(p,D) = D-1μμ(p/D) (where p is the particle momentum) and that a previously suggested model for the generation of large-scale magnetic fields through a hierarchical merger of current filaments should be generalized. A numerical experiment testing our analysis is outlined.

Nonparametric estimation of correlation functions in longitudinal and spatial data, with application to colon carcinogenesis experiments

Abstract: In longitudinal and spatial studies, observations often demonstrate strong correlations that are stationary in time or distance lags, and the times or locations of these data being sampled may not be homogeneous. We propose a nonparametric estimator of the correlation function in such data, using kernel methods. We develop a pointwise asymptotic normal distribution for the proposed estimator, when the number of subjects is fixed and the number of vectors or functions within each subject goes to infinity. Based on the asymptotic theory, we propose a weighted block bootstrapping method for making inferences about the correlation function, where the weights account for the inhomogeneity of the distribution of the times or locations. The method is applied to a data set from a colon carcinogenesis study, in which colonic crypts were sampled from a piece of colon segment from each of the 12 rats in the experiment and the expression level of p27, an important cell cycle protein, was then measured for each cell within the sampled crypts. A simulation study is also provided to illustrate the numerical performance of the proposed method.

Diffusion and segmental dynamics of rodlike molecules by fluorescence correlation spectroscopy.

Abstract: The dynamics of weakly bending polymers is analyzed on the basis of a Gaussian semiflexible chain model and the fluorescence correlation spectroscopy (FCS) correlation function is determined. Particular attention is paid to the influence of the rotational motion on the decay of the FCS correlation function. An analytical expression for the correlation function is derived, from which the averaged segmental mean square displacement can be determined independent of any specific model for the polymer dynamcis. The theoretical analysis exhibits a strong dependence of the correlation function on the rotational motion for semiflexible polymers with typical lengths and persistence lengths of actin filaments or fd viruses. Hence, FCS allows for a measurement of the rotational motion of such semiflexible polymers. The theoretical results agree well with experimental measurements on actin filaments and confirm the importance of large relaxation times.

Nonparametric estimation of correlation functions in longitudinal and spatial data, with application to colon carcinogenesis experiments

Abstract: In longitudinal and spatial studies, observations often demonstrate strong correlations that are stationary in time or distance lags, and the times or locations of these data being sampled may not be homogeneous. We propose a nonparametric estimator of the correlation function in such data, using kernel methods. We develop a pointwise asymptotic normal distribution for the proposed estimator, when the number of subjects is fixed and the number of vectors or functions within each subject goes to infinity. Based on the asymptotic theory, we propose a weighted block bootstrapping method for making inferences about the correlation function, where the weights account for the inhomogeneity of the distribution of the times or locations. The method is applied to a data set from a colon carcinogenesis study, in which colonic crypts were sampled from a piece of colon segment from each of the 12 rats in the experiment and the expression level of p27, an important cell cycle protein, was then measured for each cell within the sampled crypts. A simulation study is also provided to illustrate the numerical performance of the proposed method.

A Lagrangian-Based Approach for Determining Trajectories Taxonomy and Turbulence Regimes

Abstract: The use of the ratio between the acceleration and velocity time scales y = Ta/Tυ to separate Lagrangian trajectories in homogeneous classes is proposed. In fact, when analyzing subsurface floats data in the Atlantic Ocean and surface drifters data in the world’s ocean basins, it is observed that trajectories having different values of y are characterized by different shapes, correlation, and dispersal properties. In particular, trajectories having similar values of the acceleration and velocity time scales clearly show the influence of eddies and are characterized by an oscillating velocity correlation function. It is shown here that this trajectory screening is a useful procedure to rationalize the analysis of real Lagrangian trajectories and to avoid a mixture of different regimes, when averaging quantities. The mean statistical quantities computed averaging on quasi-homogeneous datasets put in evidence the role of the coherent structures in the dispersion properties, both in time and in the ma...

Bistatic Scattering and Emissivities of Lossy Dielectric Surfaces With Exponential Correlation Functions

Abstract: Bistatic scattering and emissivities of surfaces with exponential correlation functions are studied numerically for 2-D geometries in a numerical Maxwell model with 2-D simulations. Surfaces with exponential correlation functions are important for the active and passive microwave remote sensing of land surfaces. Because of the fine-scale features with large slopes of such surfaces, numerical accuracy, which is particularly important for the calculation of emissivity in passive remote sensing, is ensured by a variety of procedures in this paper. The rooftop function and Galerkin's method with numerical integration of near-field impedance matrix elements are used. Cubic spline interpolation is employed to connect knots on random rough surfaces. Numerical accuracy convergence tests are performed for numerical solutions of Maxwell equations by varying the number of points from 13 to 103 points per wavelength in the dielectric medium corresponding to 50-400 points per free wavelength. Surface lengths of up to 100 and 200 free wavelengths and root mean square heights of up to 0.4 and 0.8 free wavelengths, respectively, are used at 5 and 10 GHz to capture all the essential features. Because of the large number of surface unknowns (up to 80 000), the multilevel UV method is further used to accelerate the matrix equation solver. Numerical results are illustrated for both bistatic scattering and emissivities as functions of frequencies and incidence and scattering angles for cases of interests in microwave remote sensing. Comparisons are made with the second-order small perturbation method and Kirchhoff's approximation to reestablish the regimes of validity of these methods

Determining if a System is Heterogeneous: The Analysis of Single Molecule Rotational Correlation Functions and Their Limitations

Abstract: Single molecule spectroscopy can be utilized to measure distributions of individual molecular properties that may be averaged out in the ensemble measurement. For example, complex dynamics in disordered systems can be investigated by observing single molecule rotations via fluorescence spectroscopy. The rotational time of a single transient can be calculated from the correlation function of the reduced linear dichroism signal which fluctuates over time as the molecule reorients in its surroundings. Distributions of rotational time constants can be used to characterize the heterogeneity of molecular environments in the material. This paper reviews some theoretical studies on (1) the high numerical aperture effects on the final correlation function, and how it can be related to optical anisotropy decays in a bulk measurement; (2) the statistical errors resulting from the finite observation length that will propagate into distributions of rotational times. These lead to the discussions on how to interpret correctly the distribution of properties measured from a set of single molecule data, and to determine if in fact the system is heterogeneous.

Direct correlation function for the square-well potential.

Abstract: An analytical expression of direct correlation function (DCF) for the square-well potential is developed. The development is based on the first-order mean spherical approximation and its extension to the functionality of the existing radial distribution function. The developed DCF is a combination of a special polynomial function introduced in this work. The combination is piecewise in four regions [0,λ−1], [λ−1,2−λ], [2−λ,1], and [1,λ] for λ 1.5. In addition, the DCF is continuous to second-order inside hard core and discontinuous at r=1 and r=λ outside it. The behavior of DCF is analyzed by some detail calculations.

A set of novel correlation tests for nonlinear system variables

Abstract: A set of novel correlation tests using omni-directional cross-correlation functions (ODCCFs), which are based on the first order cross-correlation functions (CCF), are proposed in the present study to comprehensively detect nonlinear relationships between variables. Then the ODCCFs are combined into a set of concise formulations to provide better illustration of detected correlations and reduce the number of correlation plots. Compared to the other approaches, the new methodology brings much more power in detection of nonlinear correlations. The efficiency and effectiveness of the new algorithm are demonstrated through simulation studies and comparisons with other linear and nonlinear correlation tests. The results can be widely applied in many relevant fields.

Scaling behavior and mechanism of formation of SiO2 thin films grown by plasma-enhanced chemical vapor deposition

Abstract: This paper reports a study of the kinetic roughening of SiO2 thin films prepared by plasma-enhanced chemical vapor deposition PECVD. Tetramethylsilane has been used as a precursor, and the synthesis has been carried out under remote and in-plasma configurations. The analysis of surface topography of the films by atomic force microscopy shows an anomalous scaling behavior that cannot be represented by the FamilyVicsec scaling relation of dynamic scaling theory. The application of different methods for obtaining the roughness exponent yields different values of this exponent = 0.7 for the height-height correlation function and = 1.3 for the power spectral density function for long deposition times in all experimental conditions. Moreover, a strong variation of the exponent with deposition time has been determined for the samples grown in remote mode. This correlates with the presence of a crossover region of the growth exponent , which varies from a first value of 1.3 for low deposition times to another of 0.3 for longer deposition times. Such a variation is not found for the samples grown in the plasma, characterized by a value of 0.28. The results obtained can be explained by the combined effect in the growth process of a low diffusivity of the physisorbed species along with the existence of nonlocal interactions due to shadowing effects. These two assumptions are in agreement with the empirical knowledge existing about the kinetics of the growth of SiO2 thin films by PECVD and establish a link between the scaling properties of the films with the surface chemistry during the film growth.

Fourier transform of the 2kF luttinger liquid density correlation function with different spin and charge velocities

Abstract: We obtain a closed-form analytical expression for the zero-temperature Fourier transform of the 2kF component of the density-density correlation function in a Luttinger liquid with different spin and charge velocities. For frequencies near the spin and charge singularities, approximate analytical forms are given and compared with the exact result. We find power-law-like singularities leading to either divergence or cusps, depending on the values of the Luttinger parameters, and compute the corresponding exponents. Exact integral expressions and numerical results are given for the finite-temperature case as well. We show, in particular, how the temperature rounds the singularities in the correlation function.