Showing papers on "Correlation function (statistical mechanics) published in 1988"

The discrete correlation function: a new method for analyzing unevenly sampled variability data

Abstract: A method for measuring correlation functions without interpolating in the temporal domain is proposed which provides an assumption-free representation of the correlation measured in the data and allows meaningful error estimates. Physical interpretation of the cross-correlation function of two series believed to be related by a convolution is shown to require knowledge of the input function's fluctuation power spectrum. Application of the method to two systems reveals no correlation for the optical data of Akn 120, but a strong correlation for the UV data of NGC 4151, placing bounds of between 1.2 and 20 light days on the size of the line-emitting region.

Numerical study of the late stages of spinodal decomposition

Abstract: A numerical simulation of a two-dimensional system with a continuous order parameter (model B) is used to study phase separation for a critical quench. Domain growth and scaling are investigated through the pair correlation function. At late stages, the characteristic domain size grows as R(t)\ensuremath{\sim}${t}^{1/3}$. It is only in this time regime that scaling of the correlation function is established. The scaling function and the asymptotic growth law are found to be independent of the strength of thermal fluctuations. Details of the finite-difference scheme used to simulate the dynamics are discussed.

Critical dynamics of the sol-gel transition.

Abstract: The dynamics of the sol-gel transition is probed by use of quasielastic light scattering. A type of critical dynamics is observed that is associated with a divergent friction, rather than a singularity in a thermodynamic quantity. Several novel effects are reported, including power-law time decay of the intensity autocorrelation function, critical slowing down of the average relaxation time, and observation of a fractal time set in the scattered field.

On the universality of the two-point galaxy correlation function

Abstract: The behavior of the two-point galaxy correlation function in volume-limited subsamples of three complete redshift surveys is investigated. The correlation length is shown to scale approximately as the square root of the distance limit in both the CfA and Southern Sky catalogs, but to be independent of the distance limit in the IRAS sample. This effect is found to be due to factors such as the large positive density fluctuations in the foreground of the optically selected catalogs biasing the correlation length estimate downward, and the brightest galaxies appearing to be more strongly clustered than the mean.

Scaling properties in supercooled liquids near the glass transition

Abstract: A simple model for the liquid glass transition is solved numerically. Results for ideal and non-ideal transitions are compared. The data show two or three different scaling regions for the density correlation function or the corresponding susceptibility spectra in agreement with previous analytical results.

The Dynamics of Polar Solvation

Abstract: Fluorescence upconversion experiments with subpicosecond resolution are described. The time dependent fluorescence spectra of several probe molecules in polar solvents are measured and used to construct the Stokes shift correlation function C(t). The correlation functions decay nonexponentially in contrast to the predictions of simple Debye/Onsager continuum theory. At high values of the static dielectric constants the solvation times are much slower than predicted from standard continuum theory. Generalizations of continuum theory to include non-spherical shapes, non-Debye dielectric response and saturation effects are discussed. The experimental data are qualitatively in accord with predictions based on a frequency and position dependent dielectric response.

Constraints on the Biasing of Density Fluctuations

Abstract: Developpement en serie des fonctions de correlations d'ordre eleve du champ de densite biaise

Theory of trapped-ion-temperature-gradient-driven turbulence and transport in low-collisionality plasmas

Abstract: A novel theory for the nonlinear evolution of the trapped‐ion‐temperature‐gradient‐driven mode, based on the turbulent trapping of resonant ions in the electrostatic potential of the waves, is proposed. A statistical description is adopted whereby the self‐consistent evolution of the two‐point correlation function of the trapped‐particle distribution function is followed in phase space. Threshold‐dependent, non‐steady‐state turbulence (nonlinear instability) is shown to develop when the decay of the correlation function is overcome by a source term that derives its free energy from the relaxation of the average distribution function. This nonlinear instability leads to anomalous thermal and particle transport that in turn reconfigure the equilibrium temperature and density profiles in such a way as to return the system toward its marginal point. Expressions for the nonlinear dispersion relation and threshold, as well as estimates of the thermal and particle transport level, are derived. The estimated flux levels are sufficiently high as to make any significant departure away from marginality unlikely. The scenario outlined serves to underscore the desirability for pellet injection in experimental devices such as the Compact Ignition Tokamak [Bull. Am. Phys. Soc. 32, 1921 (1987)] that operate in the very low ion collisionality regime where this mode would be expected to become relevant. As with a number of recent theories, the present work further reinforces the notion that unfavorably drifting trapped particles pose a serious menace to confinement and suggests inboard, off‐axis radio‐frequency heating as one means of reducing the size of this population, at least for the case of those energetic trapped particles created during auxiliary heating.

The correlation functions of rbm and m/m/1

Abstract: This paper describes the (auto) correlation functions of regulated or reflecting Brownian motion (RBM) and several processes associated with the M/M/1 queue. For RBM and the M/M/1 continuous-time queue-length process, the correlation function of the stationary process coincides with the complementary stationary-excess cdf (cumulative distribution function) associated with a previously studied first-moment cdf. The first-moment cdf is the mean as a function of time given that the process starts at the origin, normalized by dividing by the steady-state limit. The M/M/1 first-moment cdf in turn is the stationary-excess cdf associated with the M/M/1 busy-period cdf. In fact, all the moment cdfs and correlation functions can be expressed directly in terms of the busy-period cdf. This structure provides the basis for simple approximations of the correlation functions and the moments as functions of time by hyperexponentials.

Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation.

Abstract: We find experimentally that in the forward direction the length scale of quasiballistic photons dominates the dynamic intensity-intensity correlation function of light multiply scattered from randomly diffusing spheres. This implies a breakdown of the photon-diffusion approximation, which is characterized by a single length, the transport mean free path. We suggest why this breakdown occurs, introduce phenomenologically a dynamic correlation length as an additional length scale, and with this extension reconcile the theory with experiment.

Rotational diffusion of a tracer colloid particle: I. Short time orientational correlations

Abstract: We extend the generalized Smoluchowski equation to descrbe the diffusional relaxation of position and orientation in a suspension of interacting spherical colloid particles. Considering a tracer particle which interacts with other particles through spherically symmetric pair potentials and with an external field we obtain a cluster expansion representation of the orientational time correlation functions for the tracer. The one and two body cluster contributions are studied explicitly at short times. Working to first order in volume fraction φ we show that the initial slope of the time correlation functions is described by a modified diffusion coefficient Dr = Dr0(1 −Crφ) where Cr is a number determined by hydrodynamic and potential interactions. We evaluate Cr numerically for spheres with slip-stick hydrodynamic boundary conditions and also for permeable spheres.

Orientation Correlation in Grain and Phase Boundaries

Abstract: The properties of grain and phase boundaries depend on five angular coordinates, i.e. three parameters specifying the orientation difference across the boundary and two parameters specifying the orientation of the boundary normal direction in space or with respect to the crystal lattice. Hence, five-dimensional boundary distribution functions have to be considered. If one considers only misorientation a three-dimensional misorientation distribution function is obtained. The deviation of this function from the #8220;uncorrelated” misorientation distribution yields the orientation correlation function. The most economical representation of these functions is the one using series expansions in terms of symmetrized harmonic functions. With the present state of experimental technique it seems to be impossible to determine the complete boundary distribution functions. However, two-dimensional analoga of these functions can be obtained from electron diffraction measurements.

Structure, thermodynamics, and orientational correlations of the nematogenic hard ellipse fluid from the Percus-Yevick equation

Abstract: The Percus-Yevick equation is solved for a fluid of hard ellipses in two dimensions. The correlation functions, including the orientation correlation function, are expanded in a set of orthogonal functions and the expansion coefficients are obtained by an iterative algorithm. Pressure and compressibility values are also determined. Orientational ordering is observed, but the isotropic-nematic phase transition observed by Vieillard-Baron (1972, J. chem. Phys., 56, 4729) is not.

Norms of bound states

Abstract: The formula for the norms of the Bethe wave functions in the form of a Jacobian plays an important role in the computation of the correlation functions. In the present paper this formula is generalized to the case of bound states.

Numerical evaluation of the statistics of acoustic scattering from a rough surface

Abstract: In this article, a formulation is given for two‐dimensional scattering of a plane wave incident on an infinitely long surface with a Gaussian distribution of surface height. The surface height may have any physically reasonable correlation function. This method generalizes previous work by the present authors, in which a finite length surface was treated numerically by Fourier methods. For each numerically generated random surface, an integral equation must be solved to find the pressure gradient on the surface. The reflection coefficients, describing the angular distribution of scattered energy, are then formed. It is shown how the various moments of the scattered pressure may be obtained efficiently from averages of these reflection coefficients. Using these techniques, numerical results are presented for the angular distribution of energy for a surface with a Gaussian correlation function of surface height and for a correlation function modeling a more realistic sea surface. Results are presented showing that it is not necessary to taper the incoming wave field in order to obtain accurate answers. For large values of the ratio of the standard deviation of height to the surface correlation length, it is found that the mode of scattering is very different to that found at small surface heights. For some cases, the maximum in the angular distribution is found in the backscatter direction, with the specular peak not at all apparent. It appears that this phenomenon is due to multiple scattering of the acoustic wave at the surface. Finally, it is shown that for the class of problems treated here, the ensemble average intensity is constant at all normal distances from the surface, while the normalized covariance of intensity varies with normal distance from the surface in a way that is, in some limiting cases, very similar to the scattering of a wave passing through a phase modulating screen or an extended random medium.

Monte Carlo study of the Potts glass with nearest-neighbour random Gaussian interaction

Abstract: The three-state Potts model with nearest-neighbour random Gaussian interaction on the simple cubic lattice is investigated by Monte Carlo simulation. Both static quantities (e.g. glass ordering susceptibility and the correlation function) and dynamic quantities (the analogue of the time-dependent Edwards-Anderson order parameter q(t) for Potts spins) are obtained. As for related models for orientational glasses, it is found that q(t) is consistent with the Kohlrausch law q(t) varies as exp(-(t/ tau )y) for a wide range of temperatures, with a strongly temperature-dependent exponent y, with y becoming very small as the temperature T to 0. The relaxation time tau increases dramatically as T is lowered; thus the system could only be equilibrated for temperatures where the correlation length is rather small. Since the critical region has not been reached, it cannot be distinguished whether the critical temperature Tc is non-zero or at T=0. If Tc=0, the divergences of tau and the ordering susceptibility are probably exponential, i.e. the system is then at its lower critical dimensionality.

A stochastic model for dispersion and concentration distribution in homogeneous turbulence

Abstract: A new approach to contaminant diffusion in homogeneous turbulence is proposed. This approach is based on solving for the Lagrangian trajectories of many particles taking into account the interaction among their velocities. The velocity field at a given instant is composed of many ‘eddies’ distributed randomly and uniformly in space. The velocity of each eddy is proportional to the cube root of its size. In this way the calculated Eulerian correlation function between any two points is consistent with observations. The present model is used to calculate concentration fluctuations, concentration averages and intermittency as functions of location and time. Results were found to be in accordance with experimental measurements. Probability distributions as functions of time and location are also calculated.

Thermodynamic properties of the classical Heisenberg chain with nearest- and next-nearest-neighbor interactions

Abstract: Exact results for the thermodynamic quantities of the one-dimensional classical Heisenberg model with nearest- and next-nearest-neighbor exchange interactions are obtained by means of the numerical transfer matrix method. For a wide range of exchange constants, the system exhibits helical short-range order on which we focus our attention. We find that the Fourier-transformed spin correlation function shows a maximum with asymmetric shape at the characteristic wave-number ±qm (≠0, ±π). The correlation length defined as the inverse of the width atq=qm obeys a simple scaling law and shows a power-law singularity at zero temperature. Results for the heat capacity and the susceptibility are also presented and discussed in connection with the helical short-range order.

Intensity correlation functions of lightwaves in a turbulent medium: an exact version of the two-scale method.

Abstract: An exact functional integral representation for the two-point intensity correlation function is first obtained by solving the moment equation and is regarded as a multiscale representation. The variable functions of integration therein involved can be effectively limited to a set of functions so that the entire phase term of the integrand becomes stationary against their arbitrary variation, exactly according to the Lagrange variational principle in dynamics. This means an approximation of the multiscale by a two-scale; thereby an exact version of the integral representations by the conventional two-scale method is obtained specifically for a collimated beam wave, including both spherical and plane waves as special cases. The result is valid independent of the medium scale that has been a basic parameter in the two-scale method and also of the medium fluctuation intensity. Nevertheless, it leads to the same zeroth-order expression as that obtained with the latter method and also to a similar first-order expression when making a formal expansion. The result is free from any expansion and is presented with a set of unperturbative equations of closed form, which can be solved iteratively. With exactly the same procedure, the three-point intensity correlation and the twofrequency intensity correlation are also obtained for the same incidence beam wave. These provide an accurate solution of the moment equation to investigate tough problems, including very large values of the scintillation index observed up to 6.

Group theoretical treatment of classical n-time correlation functions

Abstract: The methods of group theoretical statistical mechanics are used to investigate the set of n -time correlation functions in isotropic molecular liquids and in molecular liquids subjected to external electric and magnetic fields. It is found that the group theory corroborates the results of computer simulation where these are available, both for auto- and cross-correlation functions, in the field-off and field-on equilibrium conditions. Many more elements of the time correlation functions are allowed by symmetry in the presence of fields. For the first time, the symmetry and time dependence of higher-order n -time correlation functions are investigated by group theory. This results in a great simplification of the exploratory work of the computer simulator, because the group theory clearly distinguishes between those elements that vanish for all t and those that may exist in the laboratory frame ( X, Y, Z ) and the molecule-fixed frame ( x, y, z ). Group theoretical statistical mechanics reveal clearly that the statistics of a molecular liquid cannot be Gaussian in general, because three-time and higher-order correlation functions exist by symmetry. These should be observable in future computer simulation. Thus we arrive at the important conclusion that group theory applied to the dynamics of molecules in the liquid state invalidates a large number of literature theories of diffusion, including all those based on the Debye concept of “rotational” diffusion. These theories should be modified accordingly.

Correlation function study for sea ice

Abstract: For active and passive microwave remote sensing of sea ice, a correlation function of exponential form is extracted from the photograph of a horizontal thin section taken from a sample of artificially grown saline ice that closely resembled Arctic congelation sea ice. It is found that the extracted correlation lengths are consistent with the published average size of brine pockets. With the application of strong fluctuation theory and the bilocal approximation, the effective permittivity tensor is derived in the low-frequency limit for an unbounded uniaxial random medium with two-phase mixtures. Using the extracted correlation lengths, the effective permittivity tensor is computed as a function of fractional volume of brine inclusions and compared with in situ measurements at 4.8 and 9.5 GHz.

Bond orientational order in simple liquids

Abstract: The phase transition of an isotropic liquid into an anisotropic phase is described in terms of a bifurcation of the solution of an exact nonlinear integral equation for the pair distribution function. The behaviour of the correlation function of orientational fluctuations for the pair distribution function is analysed.

Correlation in the ground state of Li - , Na - and K -

Abstract: The ground-state wavefunctions for Li-, Na- and K- are calculated using the hyperspherical method. They are analysed and compared with the ground state as well as the first 1S resonance in H- to reveal the effect of correlation. In order to perform a quantitative comparison an absolute measure of angular and radial correlation based on statistical methods is used and its relevance to characterise correlation in the ground states is discussed.

On the scattering function of simple fluids in finite systems

Abstract: The density-density dynamical correlation function of a simple fluid in a finite container subject to a constant temperature difference is explicitly obtained. In small systems, such as those realized in computer experiments, new peaks appear in the scattering spectrum, even at equilibrium, arising from standing waves produced by the fluctuations. Away from equilibrium, these peaks are asymmetric in the same manner as the Brillouin lines. The macroscopic limit is also considered and the correction to the usual “infinite system approximation” is explicitly obtained.