Higher-order Correlation Functions Research Articles

A Pdf Description of Momentum Fluctuation Correlations of a Rarefied Free Shear Layer

AbstractThe mixing properties of various orders of fluctuation correlations are investigated in the present paper for a rarefied gas free shear layer at hypersonic speed. The molecular kinetic theory and the direct simulation Monte Carlo (DSMC) method are employed for the present calculations. The Reynolds average process is assumed in obtaining the correlation functions. The results show that flow field structure was very similar to that of continuum flow ones at high Reynolds numbers. The probability density functions (pdf) in velocity spacef(u′),f(v′), andf(u′,v′) are also calculated to counter explain the distributions of the correlation functions in the mixing layer. From the calculated distributions of the fluctuation correlation functions, <u′v′>, <u′2v′>, and <v′2u′>, one can find that the distributions behave similar to the turbulent transport phenomena in that of a continuum flow one. The distributions of the fluctuation correlation functions, <u′,v′> is described via the joint probability density function,f(u′,v′). The behavior of the higher-order fluctuation correlation functions, <u′2v′>, <u′3> and <u′4>, are also explained via the probability density function.

Shape of high order correlation functions in CMB anisotropy maps

We present a phenomenological investigation of non-Gaussian effects that could be seen on CMB temperature maps. Explicit expressions for the temperature correlation functions are given for different types of primordial mode couplings. We argue that a simplified description of the radial transfer function for the temperature anisotropies allows to get insights into the general properties of the bi and tri-spectra. The accuracy of these results is explored together with the use of the small scale approximation to get explicit expressions of high order spectra. The bi-spectrum is found to have alternate signs for the successive acoustic peaks. Sign patterns for the trispectra are more complicated and depend specifically on the type of metric couplings. Local primordial couplings are found to give patterns that are different from those expected from weak lensing effects.

Interference between independent fluctuating condensates.

We consider a problem of interference between two independent condensates that lack true long-range order. We show that their interference pattern contains information about correlation functions within each condensate. As an example, we analyze the interference between a pair of one-dimensional interacting Bose liquids. We find universal scaling of the average fringe contrast with system size and temperature that depends only on the Luttinger parameter. Moreover, the full distribution of the fringe contrast, which is also equivalent to the full counting statistics of the interfering atoms, changes with interaction strength and lends information on high-order correlation functions. We also demonstrate that the interference between two-dimensional condensates at finite temperature can be used as a direct probe of the Kosterlitz-Thouless transition. Finally, we discuss the generalization of our results to describe the interference of a periodic array of independent fluctuating condensates.

Pair correlations and merger bias

We analytically study the possibility that mergers of haloes are more highly clustered than the general population of haloes of comparable masses. We begin by investigating predictions for merger bias within the extended Press‐Schechter formalism and discuss the limitations and ambiguities of this approach. We then postulate that mergers occur whenever two objects form within a (small) fixed distance of each other. We therefore study the clustering of pairs of points for a highly biased population in the linear regime, for the overall mass distribution in the quasi-linear regime and (using the halo model of clustering) in the non-linear regime. Biasing, quasi-linear evolution and non-linear clustering all lead to non-zero reduced (or connected) three- and four-point correlation functions. These higher order correlation functions can in many cases enhance the clustering of close pairs of points relative to the clustering of individual points. If close pairs are likely to merge, then the clustering of mergers may be enhanced. We discuss implications for the observed clustering of luminous z = 3 galaxies and for correlations of active galactic nuclei and galaxy clusters.

Multipoint correlation functions for continuous-time random walk models of anomalous diffusion

Recursive relations are developed for computing the multipoint correlation functions of a particle undergoing a biased continuous-time random walk (CTRW) in an external potential. Two- and three-point correlation functions are calculated for waiting-time distributions with an anomalous power-law profile t(-alpha-1), 0 < alpha < 1, on intermediate time scales with a crossover to an exponential long time decay. Comparison of the CTRW with the Brownian harmonic oscillator model (Gaussian process) illustrates how higher-order correlation functions may be used to distinguish between dynamical models that have the same two-point correlation function.

Optimal estimation of non-Gaussianity

We systematically analyze the primordial non-Gaussianity estimator used by the Wilkinson Microwave Anisotropy Probe (WMAP) science team with the basic ideas of estimation theory in order to see if the limited Cosmic Microwave Background (CMB) data is being optimally utilized. The WMAP estimator is based on the implicit assumption that the CMB bispectrum, the harmonic transform of the three-point correlation function, contains all of the primordial non-Gaussianity information in a CMB map. We first demonstrate that the Signal-to-Noise (S/N) of an estimator based on CMB three-point correlation functions is significantly larger than the S/N of any estimator based on higher-order correlation functions; justifying our choice to focus on the three-point correlation function. We then conclude that the estimator based on the three-point correlation function, which was used by WMAP, is optimal, meaning it saturates the Cramer-Rao Inequality when the underlying CMB map is nearly Gaussian. We quantify this restriction by demonstrating that the suboptimal character of our estimator is proportonal to the square of the fiducial non-Gaussianity, which is already constrained to be extremely small, so we can consider the WMAP estimator to be optimal in practice. Our conclusions do not depend on the form of the primordial bispectrum, only on the observationally established weak levels of primordial non-Gaussianity.

High-order current correlation functions in the Kondo systems

We examine the statistics of current fluctuations in a junction with a quantum dot described by Kondo Hamiltonian. With the help of modified Keldish technique we calculate the third current cumulant. As a function of ratio $v=eV/T_{K}$ the 3rd cumulant was obtained for three different regimes: Fermi liquid regime (v<1), crossover interval ($v\geq1$) and RG limit (v>>1). Unlike the case of noninteracting dot, 3rd cumulant shows strong non-linear voltage dependence. Only in the asymptotical limit of large voltages the linear dependence on $V$ is recovered.

Mode-Coupling Theory for Multiple-Time Correlation Functions of Tagged Particle Densities and Dynamical Filters Designed for Glassy Systems

The theoretical framework for higher-order correlation functions involving multiple times and multiple points in a classical, many-body system developed by Van Zon and Schofield [Phys. Rev. E 2002, 65, 011106] is extended here to include tagged particle densities. Such densities have found an intriguing application as proposed measures of dynamical heterogeneities in structural glasses. The theoretical formalism is based upon projection operator techniques which are used to isolate the slow time evolution of dynamical variables by expanding the slowly evolving component of arbitrary variables in an infinite basis composed of the products of slow variables of the system. The resulting formally exact mode-coupling expressions for multiple-point and multiple-time correlation functions are made tractable by applying the so-called N-ordering method. This theory is used to derive for moderate densities the leading mode coupling expressions for indicators of relaxation type and domain relaxation, which use dynamical filters that lead to multiple-time correlations of a tagged particle density. The mode coupling expressions for higher order correlation functions are also successfully tested against simulations of a hard sphere fluid at relatively low density.

High-order 2MASS galaxy correlation functions: probing the Gaussianity of the primordial density field

We use the 2 Micron All Sky Survey (2MASS) extended source catalogue to determine area-averaged angular correlation functions, ω p , to high orders (p ≤ 9). The main sample used contains 650745 galaxies below an extinction-corrected magnitude of K s = 13.5 and limited to |b| > 10° and represents an order of magnitude increase in solid angle over previous samples used in such analysis (≈34 000 deg 2 ). The high-order correlation functions are used to determine the projected and real-space hierarchical amplitudes, s p = ω p /ω p-1 2 and S p = ξ p /ξ 2 p-1 . For p < 6 these parameters are found to be quite constant over a wide range of scales to r ≈ 40 h -1 Mpc, consistent with a Gaussian form to the primordial distribution of density fluctuations which has evolved under the action of gravitational instability. We test the sensitivity of our results to the presence of rare fluctuations in the local galaxy distribution by cutting various regions of overdensity from the main sample; unlike previous analyses, we find that at least for p ≤ 4, the high-order clustering statistics are relatively robust to the removal of the largest superclusters. Since we probe well into the linear regime we are able to make direct comparisons with perturbation theory. We therefore examine the consistency of our constraints on the K s band Sp parameters with non-Gaussian initial conditions. We are able to rule out strong non-Gaussianity in the primordial density field, as might be seeded by topological defects such as cosmic strings or global textures, producing relations of the form ξ p ∝ ξ p/2 2 , at the 2-3σ confidence level.

Minimal coupling model of the biaxial nematic phase

A minimal coupling model exhibiting isotropic, uniaxial, and biaxial nematic phases is analyzed in detail and its relation to existing models known in the literature is clarified. Its intrinsic symmetry properties are exploited to restrict the relevant ranges of coupling constants. Further on, properties of the model are thoroughly investigated by means of bifurcation theory as proposed by Kayser and Raveché [Phys. Rev. A 17, 2067 (1978)] and Mulder [Phys. Rev. A 39, 360 (1989)]. As a first step toward this goal, the bifurcation theory is applied to a general formulation of density functional theory in terms of direct correlation functions. On a general formal level, the theory is then analyzed to show that the bifurcation points from the reference, high-symmetry equilibrium phase to a low-symmetry structure depend only on the properties of the one-particle distribution function and the direct pair correlation function of the reference phase. The character of the bifurcation (whether spinodal, critical, tricritical, isolated Landau point, etc.) depends, in addition, on a few higher-order direct correlation functions. Explicit analytical results are derived for the case when only the leading L=2 terms of the potential (mean-field analysis) or of the direct pair correlation function expansion in the symmetry-adapted basis are retained. Formulas are compared with the numerical calculations for the mean-field, momentum L=2 potential model, in which case they are exact. In particular, bifurcations from the isotropic and uniaxial nematic to the biaxial nematic phases are discussed. The possibility of the recently reported nematic uniaxial-nematic biaxial tricritical point [A. M. Sonnet, E. G. Virga, and G. E. Durand, Phys. Rev. E 67, 061701 (2003)] is analyzed as well.

Information in magnetic EXAFS

To understand the information one can obtain from MEXAFS, we develop a theory of MEXAFS within the spherical wave approximation. We show that the dominant signal is directly proportional to the magnetic moment per atom and smaller than regular EXAFS by an order of magnitude. We show that this explains the observed temperature dependence of MEXAFS. The higher order correlation functions can, in principle, be determined from double scattering paths, but they give contributions which are smaller by another two orders of magnitudes compared to that from the spin moment. Thus MEXAFS can serve as an alternative method for the element specific extraction of total spin magnetic moments. This can be achieved by modification of the regular EXAFS fitting routines and the output of the spin-dependent FEFF8 code.

Non-Poisson dichotomous noise: Higher-order correlation functions and aging

We study a two-state symmetric noise, with a given waiting time distribution psi (tau) , and focus our attention on the connection between the four-time and two-time correlation functions. The transition of psi (tau) from the exponential to the nonexponential condition yields the breakdown of the usual factorization condition of high-order correlation functions, as well as the birth of aging effects. We discuss the subtle connections between these two properties and establish the condition that the Liouville-like approach has to satisfy in order to produce a correct description of the resulting diffusion process.

Non-Poisson processes: regression to equilibrium versus equilibrium correlation functions

We study the response to perturbation of non-Poisson dichotomous fluctuations that generate super-diffusion. We adopt the Liouville perspective and with it a quantum-like approach based on splitting the density distribution into a symmetric and an anti-symmetric component. To accomodate the equilibrium condition behind the stationary correlation function, we study the time evolution of the anti-symmetric component, while keeping the symmetric component at equilibrium. For any realistic form of the perturbed distribution density we expect a breakdown of the Onsager principle, namely, of the property that the subsequent regression of the perturbation to equilibrium is identical to the corresponding equilibrium correlation function. We find the directions to follow for the calculation of higher-order correlation functions, an unsettled problem, which has been addressed in the past by means of approximations yielding quite different physical effects.

Maximum entropy formulation of the Kirkwood superposition approximation.

Using a variational formulation, we derive the Kirkwood superposition approximation for systems at equilibrium in the thermodynamic limit. We define the entropy of the triplet correlation function and show that the Kirkwood closure brings the entropy to its maximal value. This approach leads to a different interpretation for the Kirkwood closure relation, usually explained by probabilistic considerations of dependence and independence of particles. The Kirkwood closure is generalized to finite volume systems at equilibrium by computing the pair correlation function in finite domains. Closure relations for high order correlation functions are also found using a variational approach. In particular, maximizing the entropy of quadruplets leads to the high order closure g(1234)=g(123)g(124)g(134)g(234)/[g(12)g(13)g(14)g(23)g(24)g(34)] used in the Born-Green-Yvon 2 equations which are a pair of integral equations for the triplet and pair correlation functions.

Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere.

To study the performance of atmospheric optical links by using Hermite-sinusoidal-Gaussian laset beam sources, we derive the log-amplitude and the phase correlation and structure functions of such beams in a turbulent atmosphere. Our formulations correctly reduce to the known higher-order mode correlation and structure functions, which in turn reduce to the fundamental-mode (TEM00-mode) results. Several special cases of our formulation are presented, among which the case involving Hermite-cosh-Gaussian dependence is especially noted, since this case is of interest to us owing to the nature of cosh dependence exhibiting the concentration of the energy in the outer lobes of the beam.

Quantum-engineered neutron states and phase space manipulations

In neutron interferometry and in spin-echo systems non-classical neutron quantum states can be produced and manipulated. Schrödinger cat-like states exhibit two spatially separated regions which are occupied by the neutron at the same time. When used in scattering experiments this enables a direct observation of the local structure and dynamics of matter. A proper description can be obtained by means of Wigner quasi-distribution functions which are common in quantum optics. More complicated quasi-distribution functions have been verified recently by means of a double loop interferometer. These quantum states have the capacity to observe even higher-order correlation functions characterizing condensed matter. The non-local character of quantum theory has also been demonstrated in the behavior of neutrons within narrow slits where typical confinement effects have been observed. Such Casimir-like and Zeno-like phenomena show how neutrons are an attractive tool for quantum optical experiments opening new possibilities of neutron phase space manipulations. A proper treatment of the neutron field can enhance the available neutron intensities considerably. First cooling to ultra-cold neutron energies by cold solid deuterium moderators increases the phase space density, which can be transferred to higher energies by a proper phase space transformer consisting of fast rotating multi-layer mirrors or crystals. Various proposals in this directions will be discussed.

Correlations in atomic systems: diagnosing coherent superpositions.

While investigating quantum correlations in atomic systems, we note that single measurements contain information about these correlations. Using a simple model of measurement-analogous to the one used in quantum optics-we show how to extract higher-order correlation functions from individual "photographs" of the atomic sample. As a possible application, we apply the method to detect a subtle phase coherence in mesoscopic superpositions.

Quantum-engineered neutron states and phase space manipulations

In neutron interferometry and in spin-echo systems non-classical neutron quantum states can be produced and manipulated. Schrödinger cat-like states exhibit two spatially separated regions which are occupied by the neutron at the same time. When used in scattering experiments this enables a direct observation of the local structure and dynamics of matter. A proper description can be obtained by means of Wigner quasi-distribution functions which are common in quantum optics. More complicated quasi-distribution functions have been verified recently by means of a double loop interferometer. These quantum states have the capacity to observe even higher-order correlation functions characterizing condensed matter. The non-local character of quantum theory has also been demonstrated in the behavior of neutrons within narrow slits where typical confinement effects have been observed. Such Casimir-like and Zeno-like phenomena show how neutrons are an attractive tool for quantum optical experiments opening new possibilities of neutron phase space manipulations. A proper treatment of the neutron field can enhance the available neutron intensities considerably. First cooling to ultra-cold neutron energies by cold solid deuterium moderators increases the phase space density, which can be transferred to higher energies by a proper phase space transformer consisting of fast rotating multi-layer mirrors or crystals. Various proposals in this directions will be discussed.

High order correlation functions for a self-interacting scalar field in de Sitter space

We present the expressions of the three- and four-point correlation functions of a self-interacting light scalar field in a de Sitter spacetime at tree order, respectively, for a cubic and a quartic potential. Exact expressions are derived and their limiting behavior on superhorizon scales are presented. Their essential features are shown to be similar to those obtained in a classical approach.

Complex lithium ion dynamics in simulatedLiPO3glass studied by means of multitime correlation functions

Molecular dynamics simulations are performed to study the lithium jumps in ${\mathrm{LiPO}}_{3}$ glass. In particular, we calculate higher-order correlation functions that probe the positions of single lithium ions at several times. Three-time correlation functions show that the nonexponential relaxation of the lithium ions results from both correlated back-and-forth jumps and the existence of dynamical heterogeneities, i.e., the presence of a broad distribution of jump rates. A quantitative analysis yields that the contribution of the dynamical heterogeneities to the nonexponential depopulation of the lithium sites increases upon cooling. Further, correlated back-and-forth jumps between neighboring sites are observed for the fast ions of the distribution, but not for the slow ions and, hence, the back-jump probability depends on the dynamical state. Four-time correlation functions indicate that an exchange between fast and slow ions takes place on the time scale of the jumps themselves; i.e., the dynamical heterogeneities are short lived. Hence, sites featuring fast and slow lithium dynamics, respectively, are intimately mixed. In addition, a backward correlation beyond the first neighbor shell for highly mobile ions and the presence of long-range dynamical heterogeneities suggest that fast ion migration occurs along preferential pathways in the glassy matrix. In the melt, we find no evidence for correlated back-and-forth motions and dynamical heterogeneities on the length scale of the next-neighbor distance.