Expression For Correlation Function Research Articles

Pair counting without binning - a new approach to correlation functions in clustering statistics

Abstract This paper presents a novel perspective on correlation functions in the clustering analysis of the large-scale structure of the universe. We begin with the recognition that pair counting in bins of radial separation is equivalent to evaluating counts-in-cells (CIC), which can be modelled using a filtered density field with a binning-window function. This insight leads to an in situ expression for the two-point correlation function (2PCF). Essentially, the core idea underlying our method is to introduce a window function to define the binning scheme, enabling pair-counting without binning. This approach develops an idea of generalised 2PCF, which extends beyond conventional discrete pair counting by accommodating non-sharp-edged window functions. In the context of multiresolution analysis, we can implement a fast algorithm to estimate the generalised 2PCF. To extend this framework to N-point correlation functions (NPCF) using current optimal edge-corrected estimators, we developed a binning scheme that is independent of the specific parameterisation of polyhedral configurations. In particular, we demonstrate a fast algorithm for the three-point correlation function (3PCF), where triplet counting is accomplished by assigning either a spherical tophat or a Gaussian filter to each vertex of triangles. Additionally, we derive analytical expressions for the 3PCF using a multipole expansion in Legendre polynomials, accounting for filtered field (binning) corrections. Our method provides an exact solution for quantifying binning effects in practical measurements and offers a high-speed algorithm, enabling high-order clustering analysis in extremely large datasets from ongoing and upcoming surveys such as Euclid, LSST, and DESI.

Loops in AdS: from the spectral representation to position space. Part III

We study loop amplitudes in anti de-Sitter space via the spectral representation. We consider loops of spinning fields and in particular gauge fields, and derive various identities connecting different families of loop diagrams, at different number of loops, different spins, different masses. Such identities are useful for the computation of Witten diagrams. Considering the theory of large-Nf conformal scalar QED defined on AdS space, we derive an analytic expression for the exact 4-point correlation function at sub-leading order in 1Nf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{N_f} $$\\end{document}. Additionally, we derive analytic expressions for bulk 2-point functions and boundary 4-point functions for various families of diagrams, which we denote as “blob diagrams”. Finally we study 4-point ladder diagrams with spinning fields, and we derive integral expressions for the spectral representation of a k-loop ladder diagram.

Correlation functions of scalar field theories from homotopy algebras

We present expressions for correlation functions of scalar field theories in perturbation theory using quantum A∞ algebras. Our expressions are highly explicit and can be used for theories both in Euclidean space and in Minkowski space including quantum mechanics. Correlation functions at a given order of perturbation theory can be calculated algebraically without using canonical quantization or the path integral, and we demonstrate it explicitly for φ3 theory. We show that the Schwinger-Dyson equations are satisfied as an immediate consequence of the form of the expressions based on quantum A∞ algebras.

Simulations and integral-equationtheories for dipolar density interacting disks.

Integral equationtheories (IETs) based on the Ornstein-Zernike (OZ) relation can be used as an analytical tool to predict structural and thermodynamic properties and phase behavior of fluids with low numerical cost. However, there are no studies of the IETs for the dipolar density interaction potential in two-dimensional systems, a relevant interdomain interaction in lipid monolayers with phase coexistence. This repulsive interaction arises due to the excess dipole density of the domains, which are aligned perpendicular to the interface. This work studies the performance of three closures of the OZ equationfor this novel system: Rogers-Young (RY), modified hypernetted chain (MHNC), and variational modified hypernetted chain (VMHNC). For the last two closures the bridge function of a reference system is required, with the hard disk being the most convenient reference system. Given that in two dimensions there is no analytical expressions for the hard disk correlation functions, two different approximations are proposed: one based on the Percus-Yevick (PY) approximation, and the other based on an extension of the hard spheres Verlet-Weis-Henderson-Grundke (LB) parametrization. The accuracy of the five approaches is evaluated by comparison of the pair correlation function and the structure factor with Monte Carlo simulation data. The results show that RY closure is satisfactory only for low-structured regimes. MHNC and VMHNC closures perform globally well, and there are no significant differences between them. However, the reference system in some cases affects their performance; when the pair correlation function serves as the measure, the LB-based closures quantitatively outperform the PY ones. From the point of view of its applicability, LB-based closures do not have a solution for all studied interaction strength parameters, and, in general, PY-based closures are numerically preferable.

Stability, structure, compressibility factor, excess free energy and density profile of hard prolate pear shape fluids

ABSTRACT In this work, the Kerr coefficients of isotropic and uniaxial nematic liquid crystals consisting of hard ellipsoid and hard prolate pear-shaped molecules were theoretically studied using classical density functional theory. A new analytical approximate expression for the direct correlation function of hard pear- shaped molecules is applied. Using the expansion coefficients of this direct correlation function, we calculated the structural parameters, compressibility factor, and excess free energy of the hard pear-shaped liquid crystals. In addition, using the restricted orientation model, one particle density of a confined hard pear- shaped liquid crystal is calculated. The limiting stabilities and estimated transition densities are in qualitative agreement with available data. The results confirm the validity of this approximate direct correlation function.

Spatial intensity correlations of transmitted intensity patterns emerging from large particles.

Propagation of a coherent light beam through a random medium generates speckle patterns, in which some information of media and object is hidden. Speckles produced by particles smaller than wavelength are studied thoroughly, yet it is also essential to investigate speckles produced by larger particles. In this paper, the spatial intensity correlations of transmitted speckle patterns generated by large particles are studied theoretically and experimentally. A semi-empirical expression of spatial intensity correlation function of speckle patterns is derived based on Bethe-Salpeter equation, taking particle size and concentration into account. After performing experiments with various particle sizes and concentrations, we fit the theoretical expression to experimental results and determine the introduced parameters. We analyze the variation of spatial intensity correlation function with particle size and concentration. Theoretical analyses and experimental results given in this paper have potential applications in coherent imaging through random and disordered media.

Korrelyatsii zavikhrennosti vnutri kogerentnogo vikhrya

We investigate fluctuations of vorticity inside a coherent vortex generated by the inverse energy cascade in two-dimensional turbulence. Temporal and spatial correlations can be characterized by the pair correlation function. The interaction of fluctuations leads to a nonzero third moment of vorticity. We analyze the pair correlation function and the third moment using a model in which the pumping is short-correlated in time and derive explicit expressions for the Gaussian spatial correlation function for the pumping force.

Minkowski Tensors in Redshift Space—Beyond the Plane-parallel Approximation

The Minkowski tensors (MTs) can be used to probe anisotropic signals in a field, and are well suited for measuring the redshift-space distortion (RSD) signal in large-scale structure catalogs. We consider how the linear RSD signal can be extracted from a field without resorting to the plane-parallel approximation. A spherically redshift-space distorted field is both anisotropic and inhomogeneous. We derive expressions for the two-point correlation functions that elucidate the inhomogeneity, and then explain how the breakdown of homogeneity impacts the volume and ensemble averages of the tensor Minkowski functionals. We construct the ensemble average of these quantities in curvilinear coordinates and show that the ensemble and volume averages can be approximately equated, but this depends on our choice of definition of the volume average of a tensor and the radial distance between the observer and field. We then extract the tensor Minkowski functionals from spherically redshift-space distorted, Gaussian random fields and gravitationally evolved dark matter density fields at z = 0 to test if we can successfully measure the Kaiser RSD signal. For the dark matter field, we find a significant, ∼10% anomalous signal in the MT component parallel to the line of sight that is present even on large scales R G ≳ 15 Mpc, in addition to the Kaiser effect. This is due to the line-of-sight component of the MT being significantly contaminated by the Finger of God effect, which can be approximately modeled by an additional damping term in the cumulants.

Spectral and cross-spectral densities expressions’ refinement for rail vehicle oscillations

In this paper, we derive analytical expressions describing spectral and mutual spectral densities of random processes by performing analytical Fourier transform of the corresponding correlation functions. On the basis of accepted analytical expressions for correlation functions of differentiable random processes, analytical expressions for spectral density and mutual spectral density components, which is a complex frequency function, are derived. The influence of the accepted analytical expressions on the correlation functions, spectral densities, and mutual spectral density components (real, imaginary, amplitude, and phase) on the parameters is presented. Plots obtained using obtained analytical expressions have showed full convergence with those obtained by direct integration of the analytical expression of the correlation function. The analytical expressions given in this paper may be used for investigating various random processes. In particular, spectral and reciprocal spectral densities can be used for approximation of experimentally received spectral densities, including equivalent geometrical irregularities of a rail track and random fluctuations of a rail vehicle. The parameters of analytical expression obtained by such an approximation can be used for generation of analogous experimental multidimensional random processes in the tasks of mathematical modelling.

Functional renormalisation group for turbulence

Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the Navier–Stokes equations, have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purposes, a sustained effort has been devoted to obtaining an effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the renormalisation group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier–Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier–Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in thisJFM Perspectivesarticle a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, we expound the derivation of the spatio-temporal behaviour of$n$-point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.

Модель спектральной корреляционной функции для сигналов с OFDM модуляцией и циклическим префиксом

The paper is aimed at building analytical models of signals possessing complex types of digital modulation within the framework of the cyclostationary approach. The proposed analytical approach includes procedures leading to closed-form analytical expressions of spectral correlation functions (SCF), which are functions depending on two arguments: frequency and cyclic frequency. They describe the probabilistic properties of the analyzed signals assuming the signals can be modeled as realizations of second-order cyclostationary random processes (CSRP). The proposed approach of obtaining both normal and conjugate SCFs is based on the modified shaping operator technique, which turns out to be an effective tool applied to CSRP analysis. The technique handles the constructing the variety of the typically used in practice CSRPs in the form of chained transforms applied to one or several independent CSRPs of known characteristics by means of relatively simple operations, widely expressed in signal and systems theory. These operations performed over signals, which are assumed to be realizations of the CSRP, correspond to the transformations of their SCF, whose formulae are presented in the paper. The paper also reveals the exact analytical expressions obtained for the normal and conjugate SCF of OFDM signals with cyclic prefix (CP) whose subcarriers are modulated using BPSK and QPSK methods. By example of OFDM signal with a CP and QPSK subcarrier modulation, a quantitative comparison was carried out between the analytical SCF assembled according to the model formula and its estimate obtained by a numerical simulation using the estimation method based on the mixed two-dimensional fast Fourier transform. It is shown that the SCF estimate converges in root-mean sense to the constructed analytical model with an increase in the duration of the analyzed signal data sample.

Channel Characterization for Hyperloops Using the Nonstationary Geometry-Based Model

As a novel means of high-speed transportation, the Hyperloop can proceed at an ultra-high speed (more than 1000 km/h) in the long and narrow pipelines. In this paper, the channel characteristic of the Hyperloop wireless communication systems is the main objective. Based on the geometric scattering theories, a novel nonstationary channel model is proposed to investigate the channel characteristics for Hyperloop train-to-ground communications. According to this model, the channel impulse response (CIR) is obtained, and the closed-form expressions of the multi-link spatial-temporal correlation functions, including the spatial cross-correlation function (CCF) and the temporal autocorrelation function (ACF) are derived and analyzed. Simulation results show that a high correlation between the multi-link channels in vacuum tube scenario can be observed. The relevant research results will contribute to the design of future Hyperloop wireless communication system.

Phase diagram of chiral magnets via Green’s function method

In this paper, Green’s function method is applied to study the ferromagnetic system with Dzyaloshinkii–Moriya (DM) interaction in both two-dimension (2D) and three-dimension chiral magnets. Relevant properties in 2D magnets are calculated, such as the susceptibility, correlation function and analytical expressions of phase boundary. Based on the theoretical results, a new phase is predicted in the window of strong DM interaction characterized by a negative winding number. In addition, helical state in pure 2D material only appears at zero temperature. The analysis on correlation function shows a special symmetry of transverse spin correlation, which corresponds to the skyrmion phase. The results also prove the instability of helical state and its lifetime is numerically computed. Different from 2D magnets, helical state in 3D exists in the window of a lower Zeeman energy and has a long lifetime. DM interaction also reduces Curie temperature because of the spatial symmetry breaking.

Coherent state representation of thermal correlation functions with applications to rate theory.

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

Soft core fluid with competing interactions at a hard wall

A model of competing pair interaction in the form of a three-Yukawa (3Y) potential is considered. The results of computer simulations show that the model proposed can describe spontaneous appearance in a homogeneous fluid of a variety of mesostructured phases. Furthermore, these self-assembly effects appear to be favored by the presence of confining walls. A field theory approach is subsequently applied to a mesoscopically homogeneous 3Y fluid at a hard wall. Explicit analytical expressions for the pair correlation function and the density profile are derived. The structure factor is calculated by introducing an effective hard core radius characteristic of the system, based on which the λ-lines is constructed. It is shown that the theoretical results obtained for density profiles of a 3Y fluid near a hard wall are in a good agreement with computer simulations data.

Slip and stress from low shear rate nonequilibrium molecular dynamics: The transient-time correlation function technique

We derive the transient-time correlation function (TTCF) expression for the computation of phase variables of inhomogenous confined atomistic fluids undergoing boundary-driven planar shear (Couette) flow at constant pressure. Using nonequilibrium molecular dynamics simulations, we then apply the TTCF formalism to the computation of the shear stress and the slip velocity for atomistic fluids at realistic low shear rates, in systems under constant pressure and constant volume. We show that, compared to direct averaging of multiple trajectories, the TTCF method dramatically improves the accuracy of the results at low shear rates and that it is suitable to investigate the tribology and rheology of atomistically detailed confined fluids at realistic flow rates.

Multipoint correlation functions at phase separation. Exact results from field theory

We consider near-critical two-dimensional statistical systems with boundary conditions inducing phase separation on the strip. By exploiting low-energy properties of two-dimensional field theories, we compute arbitrary n-point correlation of the order parameter field. Finite-size corrections and mixed correlations involving the stress tensor trace are also discussed. As an explicit illustration of the technique, we provide a closed-form expression for a three-point correlation function and illustrate the explicit form of the long-ranged interfacial fluctuations as well as their confinement within the interfacial region.

Electronic dephasing in mixed quantum–classical molecular systems using the spin-boson model

Pure electronic dephasing is investigated using the spin-boson Hamiltonian in mixed quantum–classical environment. The spin-boson model used here is a composite system made up of a quantum subsystem, an electronic 2-level subsystem linearly coupled to harmonic vibrations, interacting with a classical bath. Experimental results for a multitude of molecular systems indicate that the zero-phonon line (ZPL) profile is determined by electronic dephasing, which is not accounted for in the multimode Brownian oscillator (MBO) model due to the unphysical contribution from the MBO bath modes to the ZPL profile. Mixed quantum–classical dynamics formalism of non-equilibrium systems is employed to assess the contribution of the bath modes to pure electronic dephasing by probing the ZPL profile when coupled to a classical bath in the mixed quantum–classical condensed systems. Pure electronic dephasing is discussed in the context of mixed quantum–classical dynamics formalism which starts with mixed quantum–classical Liouville equation in a mixed quantum–classical environment. It is noteworthy, however, that the fundamental difference between the fully quantum MBO model and the mixed quantum–classical Brownian oscillator, is that the zero-phonon line calculated by the former shows unphysical asymmetry on the low-energy side as it has not been observed in real systems, whereas the ZPL reported herein eliminates this asymmetry. A systematic approach using matrix mechanics is developed to treat this phenomenon. To this end, a closed-form expression of linear and nonlinear optical electronic transition dipole moment time correlation functions in a dissipative media are derived. Linear absorption spectra and 4-wave mixing signals at various temperatures showing a sound thermal broadening, temporal decay, and accurate pure dephasing further ratify the applicability and correctness of the mixed quantum–classical dynamics approach to spectroscopy and dynamics are computed.

FZZT branes in JT gravity and topological gravity

We study Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) branes in Witten-Kontsevich topological gravity, which includes Jackiw-Teitelboim (JT) gravity as a special case. Adding FZZT branes to topological gravity corresponds to inserting determinant operators in the dual matrix integral and amounts to a certain shift of the infinitely many couplings of topological gravity. We clarify the perturbative interpretation of adding FZZT branes in the genus expansion of topological gravity in terms of a simple boundary factor and the generalized Weil-Petersson volumes. As a concrete illustration we study JT gravity in the presence of FZZT branes and discuss its relation to the deformations of the dilaton potential that give rise to conical defects. We then construct a non-perturbative formulation of FZZT branes and derive a closed expression for the general correlation function of multiple FZZT branes and multiple macroscopic loops. As an application we study the FZZT-macroscopic loop correlators in the Airy case. We observe numerically a void in the eigenvalue density due to the eigenvalue repulsion induced by FZZT-branes and also the oscillatory behavior of the spectral form factor which is expected from the picture of eigenbranes.

On-top description of the effect of excitation on electron correlation with quasiparticles

A spectrum of $\mathrm{\ensuremath{\Delta}}$-correlon quasiparticles is proposed to describe the excitation effect on electron correlation. The proposed quasiparticle description is based on the comparison of the pair-correlation functions of the excited and ground states at the special coalescence points of the configurational space. The complex form of the $\mathrm{\ensuremath{\Delta}}$-correlon wave function allows us to naturally separate regions with the excitation suppression of correlation (ESC $\mathrm{\ensuremath{\Delta}}$-correlon), where correlation is stronger in the ground state, from those with the excitation enhancement of correlation (EEC $\mathrm{\ensuremath{\Delta}}$-correlon), where correlation is stronger in the excited state. The proposed $\mathrm{\ensuremath{\Delta}}$-correlon approach is applied to characterize the important single excitations with the competition of ``anticorrelation squeezing'' and ``enhanced-correlation spreading'' of the two-electron distribution. The former feature characterizes single valence excitations of the ionic nature and this leads to the prevailing of the ESC effect in this type of excitations. In turn, the enhanced-correlation spreading characterizes single Rydberg excitations, which leads to the prevailing of the EEC effect in this type of excitations. The proposed $\mathrm{\ensuremath{\Delta}}$-correlon quasiparticles are studied both analytically and numerically. The analytical expressions for the on-top correlation functions are obtained with the two-determinantal two-electron model of the active site of an excitation. The two-dimensional contour plots of the ESC and EEC $\mathrm{\ensuremath{\Delta}}$-correlons obtained for the lowest vertical valence ionic and Rydberg excitations in the prototype molecules ${\mathrm{H}}_{2}, {\mathrm{N}}_{2}$, and ${\mathrm{C}}_{2}{\mathrm{H}}_{4}$ from the highly correlated wave functions confirm, in general, the trends in the relative correlation strength deduced from the model.