Showing papers on "Correlation function (statistical mechanics) published in 1995"

Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture I: The van Hove correlation function.

Abstract: We report the results of a large scale computer simulation of a binary supercooled Lennard-Jones liquid. We find that at low temperatures the curves for the mean squared displacement of a tagged particle for different temperatures fall onto a master curve when they are plotted versus rescaled time tD(T), where D(T) is the diffusion constant. The time range for which these curves follow the master curve is identified with the \ensuremath{\alpha}-relaxation regime of mode-coupling theory (MCT). This master curve is fitted well by a functional form suggested by MCT. In accordance with idealized MCT, D(T) shows a power-law behavior at low temperatures. The critical temperature of this power law is the same for both types of particles, and also the critical exponents are very similar. However, contrary to a prediction of MCT, these exponents are not equal to the ones determined previously for the divergence of the relaxation times of the intermediate scattering function [Phys. Rev. Lett. 73, 1376 (1994)]. At low temperatures, the van Hove correlation function (self as well as distinct part) shows almost no sign of relaxation in a time interval that extends over about three decades in time. This time interval can be interpreted as the \ensuremath{\beta}-relaxation regime of MCT. From the investigation of these correlation functions, we conclude the hopping processes are not important on the time scale of the \ensuremath{\beta} relaxation for this system and for the temperature range investigated. We test whether the factorization property predicted by MCT holds and find that this is indeed the case for all correlation functions investigated. The distance dependences of the critical amplitudes are in qualitative agreement with the ones predicted by MCT for some other mixtures. The non-Gaussian parameter for the self part of the van Hove correlation function for different temperatures follows a master curve when plotted against time t.

Design of one-dimensional chaotic maps with prescribed statistical properties

Abstract: The statistical properties of a wide class of 1D piecewise linear Markov maps are compiled. The method used enables one to address analytically the inverse problem of designing a map with a prescribed correlation function. This class of piecewise linear maps is then used as a system of reference to analyze non-Markov piecewise linear maps and to design maps with given invariant measure and correlation function.

The analysis of the widom-rowlinson model by stochastic geometric methods

Abstract: We study the continuum Widom-Rowlinson model of interpenetrating spheres. Using a new geometric representation for this system we provide a simple percolation-based proof of the phase transition. We also use this representation to formulate the problem, and prove the existence of an interfacial tension between coexisting phases. Finally, we ascribe geometric (i.e. probabilistic) significance to the correlation functions which allows us to prove the existence of a sharp correlation length in the single-phase regime.

Determinant representation for correlation functions of spin-$1/2$ $XXX$ and $XXZ$ Heisenberg magnets

Abstract: We consider zero temperature correlation functions of the spin-1/2 XXZ Heisenberg chain in the critical regime −1<Δ≦1 in a magnetic field. Starting from the algebraic Bethe Ansatz we derive representations for various correlation functions in term of determinats of Fredholm integral operators.

Determination of the Height-Height Correlation Function of Rough Surfaces from Diffuse X-Ray Scattering

Abstract: We show that the height-height correlation function of a rough surface can be determined from the diffuse x-ray scattering intensity by an explicit back-transformation without any model assumptions. This is in contrast to the conventional fitting procedure of the data with parameterized correlation functions. The method is illustrated by the example of an amorphous Zr35Co65 film evaporated on a silicon substrate with native oxide. The height-height correlation function obtained from the diffuse-scattering data is compared to the result of in situ STM measurements.

Determination of the static scaling exponent of self-affine interfaces by nonspecular x-ray scattering.

Abstract: We discuss the conditions that have to be met in a nonspecular x-ray experiment to extract the static scaling behavior of a self-affine surface independently of a special height-height correlation function chosen to model the data. It is shown that this task requires a sufficiently large parallel momentum transfer of the scattering vector. In this respect, various scattering geometries are compared, and an analytical approximation to the structure factor on the relevant range of the scattering vector is proposed. The validity of the approximation is checked numerically.

Haldane Gap and Hidden Order in the S=2 Antiferromagnetic Quantum Spin Chain

Abstract: We have investigated Haldane's conjecture for the S=2 isotropic antiferromagnetic quantum spin chain with nearest-neighbor exchange J. Using a density matrix renormalization group algorithm for chains up to L=350 spins, we find in the thermodynamic limit a finite spin gap of Delta = 0.085(5)J and a finite spin-spin correlation length xi = 49(1) lattice spacings. We establish the ground state energy per bond to be E_0=-4.761248(1)J. We show that the ground state has a hidden topological order that is revealed in a nonlocal string correlation function. This means that the physics of the S=2 chain can be captured by a valence-bond solid description. We also observe effective free spin-1 states at the ends of an open S=2 chain.

Correlation functions of the 2D sine-Gordon model

Abstract: A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.

Quantum images and critical fluctuations in the optical parametric oscillator below threshold.

Abstract: In the first part of the paper we define the spatial spectrum of squeezing following an operational procedure, and we show in general how the spatial structure of squeezed states can be described in terms of correlation functions of quadrature components of the electric field. In the second part we formulate an appropriate quantum model for a degenerate optical parametric oscillator (OPO) with plane mirrors, and we analyze it extensively below the threshold for signal generation. The correlation length diverges when the OPO approaches threshold; this picture provides an ideal completion of the analogy with critical phenomena in second-order phase transition at equilibrium. The spatial configuration of the correlation function exhibits the phenomenon of ``quantum image:'' the signal field, which is purely generated by quantum noise, shows an ordered spatial structure which is observable via the spatial correlation function.

Some Fundamental Concepts and Techniques Useful in Small-Angle Scattering Studies of Disordered Solids

Abstract: Some concepts important for the analysis of small-angle x-ray and neutron scattering data are reviewed. After the scattering cross sections are discussed, some procedures for calculating the scattered intensity are described. Special attention is devoted to the evaluation of the correlation function of a scatterer, and some important properties of this function are pointed out. Approximate expressions for the scattered intensity for qξ ≪ 1 and qξ ≫ 1 are developed from the general equations for the scattered intensity. [Here ξ is the diameter of the scatterer; q = 4πλ−1sin(θ/2);λ is the scattered wavelength; and θ is the scattering angle.] These concepts and results are applied in a review of some small-angle scattering studies of fractals and disordered solids.

Echelle Spectroscopy with a CCD at Low Signal-to-Noise Ratio

Abstract: The measurement of some physical parameters of astronomical objects can only be carried out with high resolution spectra. Unfortunately the high dispersion of the light on the detector restricts such observations to relatively bright sources. However, some spectral information can be concentrated into a single spectral “line” by a cross-correlation algorithm, allowing the observation of fainter objects. Such a technique, taken from the CORAVEL optical correlation, is presented. A complete description of the errors of the correlation function parameters is given and the minimum signal-to-noise ratio is also discussed Finally, a short investigation of the best resolution needed to observe efficiently radial velocities and velocity broadenings is made.

Improved expressions for the radial distribution function of hard spheres

Abstract: The solution of the first‐order Ornstein–Zernike equation is applied to improve the Percus–Yevick radial distribution function (RDF) of hard spheres, where the direct correlation function is postulated to hold the Yukawa form outside the hard core. Thermodynamic consistency is imposed to determine the parameters in the postulation. Very simple analytical expressions for the Laplace transform of the RDF are obtained for hard spheres and hard sphere mixtures. The resulting RDFs are compared satisfactorily with computer simulation data.

Scaling of fuzzy controllers using the cross-correlation

Abstract: The paper deals with the optimal adjustment of input scaling factors for fuzzy controllers (FCs). The method is based on the assumption that in the stationary case an optimally adjusted input scaling factor meets a specific statistical input output dependence. A measure for the strength of statistical dependence is the correlation function and the correlation coefficient, respectively. Without loss of generality, the adjustment of input scaling factors using correlation functions is pointed out by means of a single input-single output (SISO)-system. First, the paper deals with the so-called equivalent gain which is closely connected to the cross-correlation of the controller input and the defuzzified controller output. Next, it considers the computation of correlation functions and their representation inside the FC. The paper concludes with an example of a system of fuzzy rules controlling a redundant robot manipulator. >

Magnetic X-Ray Dichroism Study of the Nearest-Neighbor Spin-Spin Correlation Function and Long-Range Magnetic Order Parameter in Antiferromagnetic NiO

Abstract: A new effect on the X-ray absorption line shape is described which is proportional to the nearest-neighbor spin-spin correlation function. We present theory and demonstrate the use of linear polarized x-ray absorption spectroscopy to study the temperature dependence of the long-range order parameter and the nearest-neighbor spin-spin correlation function in antiferro-magnetic NiO.

A Real-Space Approach to Electronic Transport

Abstract: Using orthogonal polynomials, a novel approach for studying DC and AC conductivity and velocity-velocity correlation function has been developed. The method works in direct space and can treat order or disordered, finite or infinite, and pure or alloy systems with equal ease. Further, it is not computer intensive and allows conductivity calculations as a function of frequency or the location of the Fermi energy in an efficient manner.

Molecular dynamics simulations of a Gay–Berne nematic liquid crystal: Elastic properties from direct correlation functions

Abstract: We report molecular dynamics simulations of a Gay–Berne nematic liquid crystal at constant temperature and density/pressure using the generalization of an algorithm recently proposed by Toxvaerd [Phys. Rev. E 47, 343 (1993)]. On the basis of these simulations the absolute values of the Oseen–Zocher–Frank elastic constants K11, K22, and K33 as well as the surface constants K13 and K24 have been calculated ab initio within the framework of the direct correlation function approach of Lipkin et al. [J. Chem. Phys. 82, 472 (1985)]. The angular coefficients of the direct pair correlation function, which enter the final equations, have been determined from the computer simulation data for the pair correlation function of the nematic by combining the Ornstein–Zernike relation and the Wiener–Hopf factorization scheme. The unoriented nematic approximation has been assumed when constructing the reference state of Lipkin et al. By an extensive study of the model over a wide range of temperatures, densities and pressures, very detailed information is provided on the elastic behavior of the Gay–Berne nematic. Interestingly, it is found that the results for the surface elastic constants are qualitatively different from those obtained with the help of analytical approximations for the isotropic direct pair correlation function. For example, the values of the surface elastic constants are partly negative and an order of magnitude smaller than the bulk elasticity. The negative values of the surface constant K13 indicate on the possibility of surface instabilities of the director pattern in a thin, free standing or weakly anchored Gay–Berne nematic liquid crystal.

Particle-cluster aggregation with dipolar interactions.

Abstract: The properties of aggregates generated from an off-lattice, two-dimensional, particle-cluster aggregation model with dipolar interparticle interactions have been investigated. The fractal dimension seems to be a monotonically decreasing function of the temperature, between a definite value close to 1 at T=0 and the limit T\ensuremath{\rightarrow}\ensuremath{\infty}, corresponding to diffusion-limited aggregation of particles with no interaction. Temperature and dipolar interactions are introduced by means of a Metropolis algorithm. By analyzing the orientational correlation function, and what we call the orientation probability density of the direction of the dipoles on the clusters, an ordered state is found at low temperatures. This order diminishes when the temperature increases, due to the disorder induced by the fractal geometry of the aggregates. Our study is extended to the three-dimensional case. A disordered state is found even at T=0, and a remnant order is shown when analyzing related two-dimensional correlations.

Theory of universal fast orientational dynamics in the isotropic phase of liquid crystals

Abstract: A theoretical treatment is presented that demonstrates universal dynamical behavior in the isotropic phase of liquid crystals on ultrafast time scales and short distance scales. The theoretical development generates a temperature independent power law for the short time scale decay of the molecular orientational correlation function. This provides a theoretical rationale for the postulate of universal behavior based on recent experimental observations on two liquid crystal systems. A temperature independent power law decay with the identical exponent, 0.63, was observed for the two systems. First, an alternative theoretical approach reproduces the Landau de Gennes results for the long distance scale, slow time scale orientational dynamics in the isotropic phase. This approach is also capable of examining the short distance scale and short time scale dynamics, and yields a temperature independent power law decay with exponent 0.5. Then critical correlations of fluctuations and local symmetry considerations...

Three-Point Correlations in the COBE DMR 2 Year Anisotropy Maps

Abstract: We compute the three-point temperature correlation function of the COBE Differential Microwave Radiometer (DMR) 2 year sky maps to search for evidence of non-Gaussian temperature fluctuations. We detect three-point correlations in our sky with a substantially higher signal-to-noise ratio than from the first-year data. However, the magnitude of the signal is consistent with the level of cosmic variance expected from Gaussian fluctuations, even when the low-order multipole moments, up to l = 9, are filtered from the data. These results do not strongly constrain most existing models of structure formation, but the absence of intrinsic three-point correlations on large angular scales is an important consistency test for such models.

Bringing the cross-correlation method up to date

Abstract: The cross-correlation (XC) method of Tonry & Davis (1979, AJ, 84, 1511) is generalized to arbitrary parametrized line profiles. In the new algorithm the correlation function itself, rather than the observed galaxy spectrum, is fitted by the model line profile: this removes much of the complication in the error analysis caused by template mismatch. Like the Fourier correlation quotient (FCQ) method of Bender (1990, A&A, 229, 441), the inferred line profiles are, up to a normalization constant, independent of template mismatch as long as there are no blended lines. The standard reduced chi(exp 2) is a good measure of the fit of the inferred velocity distribution, largely decoupled from the fit of the spectral template. The updated XC method performs as well as other recently developed methods, with the added virtue of conceptual simplicity.

Glassy solutions of the Kardar-Parisi-Zhang equation.

Abstract: It is shown that the mode-coupling equations for the strong-coupling limit of the KPZ equation have a solution for d>4 such that the dynamic exponent z is 2 (with possible logarithmic corrections) and that there is a delta function term in the height correlation function = (A/k^{d+4-z}) \delta(w/k^z) where the amplitude A vanishes as d -> 4. The delta function term implies that some features of the growing surface h(x,t) will persist to all times, as in a glassy state.