Third-order Correlation Research Articles

Natural scene statistics relate to perceptual salience of second-, third-, and fourth-order spatial correlations

The statistical regularities of natural scenes are a starting point for understanding the characteristics of early visual processing, e.g. the center-surround architecture of retinal ganglion cell receptive fields. Can this matching between natural signal statistics and neural processing mechanisms be extended beyond the sensory periphery? Our recent work [1] revealed that human sensitivity to fourth-order correlations in synthetic textures, known to arise in cortex, is closely related to the structure of fourth-order spatial correlations in natural scenes. This leads us to propose a specific organizing principle: The perceptual salience of visual textures increases with the variance, or unpredictability, of the corresponding correlations over the ensemble of natural scenes. To test this principle, we examined the statistical regularities of binarized natural images as characterized by correlations between adjacent pixels within a 2 × 2 square. Local binary textures can be described by four types of second-order correlations between pixels arranged in vertical (βV), horizontal (βH) and diagonal (β and β/) configurations, four types of third-order correlations between pixel triplets (θ1,2,3,4), and one type of fourth-order correlation between pixel quadruplets (α). We measured the values of these correlations in a large ensemble of image patches, and we compared the results to psychophysical experiments that measure human sensitivity to synthetic visual textures. Both the ordering and magnitude of natural image variances was found to match perceptual sensitivities to synthetic textures generated with the corresponding correlations (Figure ​(Figure1a).1a). Furthermore, the principal components of the full 9D space of image statistics match the principal components of the corresponding space of sensitivities (Figure 1b-e). These results suggest that central neural mechanisms are efficiently tuned to the higher-order statistics of natural scenes. Figure 1 (a) Variances in image statistics match perceptual sensitivities. Principal components of image statistics match principal components of human sensitivities, as shown by the fractional contribution of (b) βHV, (c) β/\, (d) θ, and ...

Role of intensity fluctuations in third-order correlation double-slit interference of thermal light

A third-order double-slit interference experiment with a pseudothermal light source in the high-intensity limit has been performed by actually recording the intensities in three optical paths. It is shown that not only can the visibility be dramatically enhanced compared to the second-order case as previously theoretically predicted and shown experimentally, but also that the higher visibility is a consequence of the contribution of third-order correlation interaction terms, which is equal to the sum of all contributions from second-order correlation. It is interesting that, when the two reference detectors are scanned in opposite directions, negative values for the third-order correlation term of the intensity fluctuations may appear. The phenomenon can be completely explained by the theory of classical statistical optics and is the first concrete demonstration of the influence of the third-order correlation terms.

Third-order statistics and the dynamics of strongly anisotropic turbulent flows

Anisotropy is induced by body forces and/or mean large-scale gradients in turbulent flows. For flows without energy production, the dynamics of second-order velocity or second-order vorticity statistics are essentially governed by triple correlations, which are at the origin of the anisotropy that penetrates towards the inertial range, deeply altering the cascade and the eventual dissipation process, with a series of consequences on the evolution of homogeneous turbulence statistics: in the case of rotating turbulence, the anisotropic spectral transfer slaves the multiscale anisotropic energy distribution; nonlinear dynamics are responsible for the linear growth in terms of Ωt of axial integral length-scales; third-order structure functions, derived from velocity triple correlations, exhibit a significant departure from the 4/5 Kolmogorov law. We describe all these implications in detail, starting from the dynamical equations of velocity statistics in Fourier space, which yield third-order correlations at three points (triads) and allow the explicit removal of pressure fluctuations. We first extend the formalism to anisotropic rotating turbulence with ‘production’, in the presence of mean velocity gradients in the rotating frame. Second, we compare the spectral approach at three points to the two-point approach directly performed in physical space, in which we consider the transport of the scalar second-order structure function ⟨(δq)2⟩. This calls into play componental third-order correlations ⟨(δq)2δu⟩(r) in axisymmetric turbulence. This permits to discuss inhomogeneous anisotropic effects from spatial decay, shear, or production, as in the central region of a rotating round jet. We show that the above-mentioned important statistical quantities can be estimated from experimental planar particle image velocimetry, and that explicit passage relations systematically exist between one- and two-point statistics in physical and spectral space for second-order tensors, but also sometimes for third-order tensors that are involved in the dynamics.

Noninvasiveness and time symmetry of weak measurements

Measurements in classical and quantum physics are described in fundamentally different ways. Nevertheless, one can formally define similar measurement procedures with respect to the disturbance they cause. Obviously, strong measurements, both classical and quantum, are invasive—they disturb the measured system. We show that it is possible to define general weak measurements, which are noninvasive: the disturbance becomes negligible as the measurement strength goes to zero. Classical intuition suggests that noninvasive measurements should be time symmetric (if the system dynamics is reversible) and we confirm that correlations are time-reversal symmetric in the classical case. However, quantum weak measurements—defined analogously to their classical counterparts—can be noninvasive but not time symmetric. We present a simple example of measurements on a two-level system which violates time symmetry and propose an experiment with quantum dots to measure the time-symmetry violation in a third-order current correlation function.

Age and isokinetic peak torque at the elbow in young girl swimmers

OBJECTIVE: This cross-sectional study examined the relationships between age and absolute peak torque (PT) as well as PT covaried for body weight (BW), height (HT), and/or fat-free weight (FFW) in young girl swimmers. METHODS: Twenty-nine girl swimmers (age = 12.4 ± 1.9 yr) were measured for isokinetic elbow flexion and extension PT at 30, 180, 300 ◦ · s −1 , as well as BW, HT, and FFW. Zero-order correlations as well as first, second, and third-order partial correlations were used to determine the relationships for age versus PT and age versus PT covaried for BW, HT, and/or FFW. RESULTS:There were age-related increases for all absolute PT measures (r = 0.47 to 0.77), except elbow flexion at 300 ◦ · s −1 . For all but one of the measures (forearm extension at 30 ◦ · s −1 ; r = 0.42 to 0.56), the increases in PT were accounted for by changes in BW, HT, and/or FFW. CONCLUSION: The results of this study indicated that agef-related increases in PT for four of the six measures were accounted for by BW, HT, and/or FFW. The age-related increases in PT for elbow flexion at 30 ◦ · s −1 independent of BW, HT, and/or FFW,

MP2.5 and MP2.X: Approaching CCSD(T) Quality Description of Noncovalent Interaction at the Cost of a Single CCSD Iteration

The performance of the second-order Møller-Plesset perturbation theory MP2.5 and MP2.X methods, tested on the S22, S66, X40, and other benchmark datasets is briefly reviewed. It is found that both methods produce highly accurate binding energies for the complexes contained in these data sets. Both methods also provide reliable potential energy curves for the complexes in the S66 set. Among the routinely used wavefunction methods, the only other technique that consistently produces lower errors, both for stabilization energies and geometry scans, is the spin-component-scaled coupled-clusters method covering iterative single- and double-electron excitations, which is, however, substantially more computationally intensive. The structures originated from full geometrical gradient optimizations at the MP2.5 and MP2.X level of theory were confirmed to be the closest to the CCSD(T)/CBS (coupled clusters covering iterative single- and double-electron excitations and perturbative triple-electron excitations performed at the complete basis set limit) geometries among all the tested methods (e.g. MP3, SCS(MI)-MP2, MP2, M06-2X, and DFT-D method evaluated with the TPSS functional). The MP2.5 geometries for the tested complexes deviate from the references almost negligibly. Inclusion of the scaled third-order correlation energy results in a substantial improvement of the ability to accurately describe noncovalent interactions. The results shown here serve to support the notion that MP2.5 and MP2.X are reasonable alternative methods for benchmark calculations in cases where system size or (lack of) computational resources preclude the use of CCSD(T)/CBS computations. MP2.X allows for the use of smaller basis sets (i.e. 6-31G*) with results that are nearly identical to those of MP2.5 with larger basis sets, which dramatically decreases computation times and makes calculations on much larger systems possible.

The role of directionality on the structure and dynamics of strongly anisotropic turbulent flows

Strong anisotropy in turbulent flows may be induced by body forces, Coriolis, buoyancy, Lorentz, and/or by large-scale gradients. These effects combined to the redistribution pressure terms are first identified by an angle dependence of the wave vector in Fourier space, the directionality. The resulting anisotropic structure is not taken into account in classical phenomenological theory, using essentially ‘isotropised’ dimensional analysis. Besides, it is generally hidden in practical engineering models by means of tuned constants, which may vary if the flow changes of nature. In this paper, different examples of anisotropic turbulence are revisited and compared to each other in order to shed light on fundamental aspects of this specific turbulence. To begin with, flows without energy production like rotating turbulence are considered. In this case, isotropy is broken by mean of third-order correlations in the equations. These correlations quantify the interscale energy transfer, and must be investigated at three-point, or triad by triad in Fourier space. This allows to account for the role of typical anisotropic frequency 2Ωcos θ k , with θ k the angle of the wave vector to the axis of rotation, and to simultaneously restore the role of phase coherence. We pursue the discussion with a second flow case, with production, quasi-static magnetohydrodynamics. This illustrates turbulence forced towards two-dimensional structure by an explicit Ohmic dissipation term. Linear dynamics displays an angle (called Moreau, or Shebalin) capable of reflecting the basic anisotropy in models as simple as . In the final phase of transition towards 2D structure, however, dynamics are essentially driven by third-order velocity correlations, and both successive linear and nonlinear phases yield counter-intuitive anisotropic results. The last case considered here is the turbulent mixing induced by a Rayleigh-Taylor instability. It is shown that anisotropy plays a central role in the dynamics of the mixing zone by means of an angular dimensionality parameter similar to the Moreau angle but for the density field, and appearing in a global model of buoyancy-drag equation.

Third-order spatial correlations for ultracold atoms

We present here the first measurement of the third-order spatial correlation function for atoms, made possible by cooling a metastable helium cloud to create an ultracold thermal ensemble just above the Bose–Einstein condensation point. The resulting large correlation length well exceeds the spatial resolution limit of the single-atom detection system, and enables extension of our earlier temporal measurements to evaluate the third-order correlation function in the spatial plane of the detector. The enhancement of the spatial third-order correlation function above a value of unity demonstrates the presence of spatial three-atom bunching, as expected for an incoherent source.

Numerical model of a turbulent flow behind a heated grid in a wind tunnel

A closure of the Corrsin equation is performed using the gradient hypothesis connecting a third-order mixed correlation moment with a second-order two-point correlation function of a passive scalar field. A numerical model of the locally isotropic turbulence is constructed based on the closed system of the Kolmogorov and Yaglom equations. On the assumption of constant Loitsiansky and Corrsin invariants, a self-similar solution of the Corrsin equation is constructed corresponding to infinitely large Reynolds and Peclet numbers. A numerical model of the turbulence dynamics and temperature fluctuations behind a heated grid in a wind tunnel is constructed based on the closed Karman-Howarth and Corrsin equations.

Von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

AbstractWe argue that the hypothesis of preservation of shape of dimensionless second- and third-order correlations during decay of incompressible homogeneous magnetohydrodynamic (MHD) turbulence requires, in general, at least two independent similarity length scales. These are associated with the two Elsässer energies. The existence of similarity solutions for the decay of turbulence with varying cross-helicity implies that these length scales cannot remain in proportion, opening the possibility for a wide variety of decay behaviour, in contrast to the simpler classic hydrodynamics case. Although the evolution equations for the second-order correlations lack explicit dependence on either the mean magnetic field or the magnetic helicity, there is inherent implicit dependence on these (and other) quantities through the third-order correlations. The self-similar inertial range, a subclass of the general similarity case, inherits this complexity so that a single universal energy spectral law cannot be anticipated, even though the same pair of third-order laws holds for arbitrary cross-helicity and magnetic helicity. The straightforward notion of universality associated with Kolmogorov theory in hydrodynamics therefore requires careful generalization and reformulation in MHD.

Resolution enhancement of third-order thermal light ghost imaging in the photon counting regime

In a three-photon spatial correlation measurement in which we scan two of three detectors in opposite directions, we observed the enhancement of both the resolution and the contrast of the third-order spatial correlation function of chaotic thermal light in the single photon counting regime. The effect can be understood as the resolution and contrast enhancement of the point-to-point correlation function that connects the imaging plane and the object plane.

Relating statistics to dynamics in axisymmetric homogeneous turbulence

The structure and the dynamics of homogeneous turbulence are modified by the presence of body forces such that the Coriolis or the buoyancy forces, which may render a wide range of turbulence scales anisotropic. The corresponding statistical characterization of such effects is done in physical space using structure functions, as well as in spectral space with spectra of two-point correlations, providing two complementary viewpoints. In this framework, second-order and third-order structure functions are put in parallel with spectra of two-point second- and third-order velocity correlation functions, using passage relations. Such relations apply in the isotropic case, or for isotropically averaged statistics, which, however, do not reflect the actual more complex structure of anisotropic turbulence submitted to rotation or stratification. This complexity is demonstrated in this paper by orientation-dependent energy and energy transfer spectra produced in both cases by means of a two-point statistical model for axisymmetric turbulence. We show that, to date, the anisotropic formalism used in the spectral transfer statistics is especially well-suited to analyze the refined dynamics of anisotropic homogeneous turbulence, and that it can help in the analysis of isotropically computed third-order structure function statistics often used to characterize anisotropic contexts.

High-Order Correlated Motion in Single-Molecule Human Adenylate Kinase 1

The growing body of data shows protein motion plays an important role in the catalytic processes of enzymes. Human adenylate kinase 1 (AK1) is the key enzyme in maintaining the cellular energy homeostasis. The conformational change of AK1 involves large-amplitude rearrangements of the enzyme's lid domain. Observing a single molecule can remove the ensemble average, thus allows the exploration of hidden structural heterogeneity. To observe the enzyme actions of AK1 at the single-molecule level, we developed a data-taking scheme allowing us to observe the single-molecule kinetics in long time without the limitation by photobleaching. We labeled the core domain and the rim of AK with two Alexa-532 chromophores. Due to the self quenching from photo-induced electron transfer, the conformation changes of the core domain can be accurately encoded into the fluorescent intensity traces with a spatial sensitivity <1nm. Because Mg2+ ion is known to be an affecter in the energy signaling network in cells, we first investigated the effect of Mg2+ on the catalytic kinetics of AK. By analyzing the photon traces, we found that the binding of Mg2+ ions to arginines and lysines of AK increases the structural heterogeneity, which then couples to the core domain and changes the catalytic function. We further discovered that ATP/AMP substrate binding suppresses the conformation fluctuation of AK and improves the thermal stability of the core domain. The resulting third-order correlation functions of photon traces do not have time-reversal symmetry, indicating single-molecule AK is in a nonequilibrium steady state. We employed hidden Markov model with Baysian inference to verify that the photon traces are originated from transitions among three states (open, mid, close). Substrate binding to AK causes different domains of AK to move in a sequential order and suppresses the thermally-activated random motions.

Bounds and correlation approximation for the effective conductivity of heterogeneous plates

The size effect obtained when studying the effective properties of plates is investigated by producing a third-order correlation approximation and Hashin-Shtrikman bounds for the effective in-plane conductivity of heterogeneous plates. The boundary condition of the plates is taken into account by obtaining the exact Green operator for the boundary problem. Results are obtained for a two-dimensional (2D) random distribution of disks and a 3D distribution of spheres. All results recover those obtained for an infinite medium when the heterogeneity size becomes small compared to plate thickness. They show that the size effect is more significant in the case of a 2D distribution of heterogeneities than for a 3D distribution.

Three-Photon Correlations in a Strongly Driven Atom-Cavity System

The quantum dynamics of a strongly driven, strongly coupled single-atom-cavity system is studied by evaluating time-dependent second- and third-order correlations of the emitted photons. The coherent energy exchange, first, between the atom and the cavity mode, and second, between the atom-cavity system and the driving laser, is observed. Three-photon detections show an asymmetry in time, a consequence of the breakdown of detailed balance. The results are in good agreement with theory and are a first step towards the control of a quantum trajectory at larger driving strength.

Third-order corrections to random-phase approximation correlation energies

Several random-phase approximation (RPA) correlation methods were compared in third order of perturbation theory. While all of the considered approaches are exact in second order of perturbation theory, it is found that their corresponding third-order correlation energy contributions strongly differ from the exact third-order correlation energy contribution due to missing interactions of the particle-particle-hole-hole type. Thus a simple correction method is derived which makes the different RPA methods also exact to third-order of perturbation theory. By studying the reaction energies of 16 chemical reactions for 21 small organic molecules and intermolecular interaction energies of 23 intermolecular complexes comprising weakly bound and hydrogen-bridged systems, it is found that the third-order correlation energy correction considerably improves the accuracy of RPA methods if compared to coupled-cluster singles doubles with perturbative triples as a reference.

An incorporative statistic and neural approach for crop yield modelling and forecasting

An incorporative framework is proposed in this study for crop yield modelling and forecasting. It is a complementary approach to traditional time series analysis on modelling and forecasting by treating crop yield and associated factors as a non-temporal collection. Statistics are used to identify the highly related factor(s) among many associates to crop yield and then play a key role in data cleaning and a supporting role in data expansion, if necessary, for neural network training and testing. Wheat yield and associated plantation area, rainfall and temperature in Queensland of Australia over 100 years are used to test this incorporative approach. The results show that well-trained multilayer perceptron models can simulate the wheat production through given plantation areas with a mean absolute error (MAE) of ~2%, whereas the third-order polynomial correlation returns an MAE of ~20%. However, statistical analysis plays a key role in identifying the most related factor, detecting outliers, determining the general trend of wheat yield with respect to plantation area and supporting data expansion for neural network training and testing. The combination of these two methods provides both meaningful qualitative and accurate quantitative data analysis and forecasting. This incorporative approach can also be useful in data modelling and forecasting in other applications due to its generic nature.

Wall-Wake Flows Downstream of a Sphere Placed on a Plane Rough Wall

The time-averaged characteristics of turbulent wall-wake flows downstream of a sphere placed on a rough wall are studied. The profiles of the defect of streamwise velocity, Reynolds shear stress, and turbulence intensities exhibit some degree of similarities when they are scaled by their respective peak defect values. For the velocity defect profiles, the vertical distances are scaled by the height of the location of the half-peak velocity defect. However, for the defect profiles of the Reynolds shear stress and the turbulence intensities, the vertical distances are scaled by the height of the location of the half-peak Reynolds shear stress defect. The magnitudes of the peak defect of all the quantities diminish with the distance downstream of the sphere characterizing the recovery of their undisturbed profiles. Additionally, the theoretical similarity solution for the velocity defect profiles is obtained. The third-order correlations imply that in the inner layer of wall wakes, a streamwise acceleration ...

Three-photon states in nonlinear crystal superlattices

It has been a longstanding goal in quantum optics to realize controllable sources generating joint multiphoton states, particularly, photon triplet with arbitrary spectral characteristics. We demonstrate that such sources can be realized via cascaded parametric down-conversion (PDC) in superlattice structures of nonlinear and linear segments. We consider scheme that involves two parametric processes: $\omega_{0}\rightarrow\omega_{1}+\omega_{2}$, $\omega_{2}\rightarrow\omega_{1}+\omega_{1}$ under pulsed pump and investigate spontaneous creation of photon triplet as well as generation of high-intensity mode in intracavity three-photon splitting. We show preparation of Greenberger-Horne-Zeilinger polarization entangled states in cascaded type-II and type-I PDC in framework of consideration dual-grid structure that involves two periodically-poled crystals. We demonstrate the method of compensation of the dispersive effects in non-linear segments by appropriately chosen linear dispersive segments of superlattice for preparation heralded joint states of two polarized photons. In the case of intracavity three-photon splitting, we concentrate on investigation of photon-number distributions, third-order photon-number correlation function as well as the Wigner functions. These quantities are observed both for short interaction time intervals and in over transient regime, when dissipative effects are essential.

Near-Bed Turbulence Characteristics at the Entrainment Threshold of Sediment Beds

This experimental study is devoted to quantification of the near-bed turbulence characteristics at an entrainment threshold of noncohesive sediments. Near the bed, the departure in the distributions of the observed time-averaged streamwise velocity from the logarithmic law is more for immobile beds than for entrainment-threshold beds. In the Reynolds shear stress distributions, a damping that occurs near the bed for sediment entrainment is higher than that for immobile beds. Quadrant analysis reveals that in the near-bed flow zone, ejections and sweeps on immobile beds cancel each other, giving rise to the outward interactions, whereas sweeps are the dominant mechanism toward sediment entrainment. The bursting duration for entrainment-threshold beds is smaller than that for immobile beds. On the other hand, the bursting frequency for entrainment-threshold beds is larger than that for immobile beds. The third-order correlations indicate that during sediment entrainment, a streamwise acceleration associated with a downward flux and advection of streamwise Reynolds normal stress is prevalent. The streamwise and the downward vertical fluxes of turbulent kinetic energy (TKE) increase with sediment entrainment. The TKE budget proves that for sediment entrainment, the pressure energy diffusion changes drastically to a negative magnitude, indicating a gain in turbulence production.