Correlation Functions Of Fields Research Articles

Rudin extension theorems on product spaces, turning bands, and random fields on balls cross time

Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into d-dimensional Euclidean spaces, Rd, have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggests that the problem is challenging. This paper provides extension theorems for multiradial characteristic functions that are defined in balls embedded in Rd cross, either Rd′ or the unit sphere Sd′ embedded in Rd′+1, for any two positive integers d and d′. We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.

Statistical Modeling of Extreme Precipitation in Summer in the Baikal Region with the Use of the Correlation Theory of Random Fields

A probabilistic model of the sums of extreme precipitation was developed for heavy showers covering large areas in the Baikal region. The spatial correlation functions of precipitation fields over one day and the entire rain period were studied. Data on the Iya River basin were used to evaluate the errors in estimating the mean values over a specified contour of extreme precipitation. Errors in the interpolation of precipitation estimates for the Baikal region were evaluated in the absence of observation data with estimation of the errors of the obtained values by Drozdov–Shepelevskii formulas. These errors amounted to about 10–15% and more. The specific features of precipitation field structures were studied with the use of their expansion in series in natural orthogonal functions for one day and over a rain period for different samples: for the entire combination of cases and for samples containing 10 and 30 maximal precipitation totals for each weather station, i.e., for extreme events. It was found that, if the observation data are limited to a range of maximal values, the structure of precipitation fields is simplified and the first 4–5 decomposition components are enough for its description. The obtained results are of importance for forecast problems and for the construction of simulation models of precipitation fields for further use in deterministic runoff simulation.

Quantum Fluctuations in the Small Fabry–Perot Interferometer

Spectra of the small Fabry–Perot interferometer (FPI) of the size of the order of the wavelength, with the main mode excited by a quantum field from a nano–LED or a laser, are investigated. The input field is detuned from the FPI mode with only a few photons. We formulate the convenient model for the FPI interacting with a quantum field, and provide novel explicit expressions for the field and the photon number fluctuation spectra inside and outside the FPI, with clearly identified contributions of the quantum and the classical noise. As a result, we found the spectra structures are quite different for the field, the photon number fluctuations inside the FPI, for the transmitted and the reflected fields and note asymmetries in spectra. The quantum noise is colored (or white) inside (or outside) the FPI, which explains differences in spectra. As another novel result, we calculate the second-order time auto–correlation functions for the FPI field; they oscillate and are negative under certain conditions. Results will help the study, design, manufacture, and use of the small elements of quantum optical integrated circuits, such as delay lines or optical transistors.

The evaluation of the interaction potential of two non-polar molecules in Green function approach

In this research, by assuming that the vacuum electric field fluctuations can convert a molecule (or an atom) into an oscillating electric dipole, the interaction potential of the two atoms or two non-polar molecules is calculated in terms of the molecular polarizabilities and the distance between them. Considering the net electric field at the position of a molecule as a sum of the vacuum electric field and the electric field due to the induced electric dipole of the other molecule, and substituting in the quadratic stark shift formula, the interaction potential of the two molecules is related to the vacuum electric field correlation function. By writing the interaction potential in terms of the imaginary parts of vector potential Green function tensor components (via the fluctuation- dissipation theorem and Kubo’s formula in statistical mechanics) and computing the required Green function components, the interaction potential between the two molecules is evaluated. The small and large distances limits of the general formula investigated and the consistencies with the previous works are shown.‎

Modified Klyshko method for an analog detector calibration.

We extend the absolute quantum efficiency (QE) calibration method, previously used only for photon counters, to analog detectors with a high dispersion of single-photon responses. Our approach is demonstrated on a reference-free measurement of a photomultiplier tube (PMT) cathode QE with 4% relative uncertainty. It involves not only measuring the biphoton field correlation function, but also a special approximation of the distribution of detector readings in order to determine the average number of photoelectrons and the average charge corresponding to a single photoelectron. Results of the calibration were verified in an independent experiment; the numbers of incident photons detected by the PMT and a single-photon counter are in good agreement.

Conformal Field Theory at the Lattice Level: Discrete Complex Analysis and Virasoro Structure

Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333–380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question (Itoyama and Thacker in Phys Rev Lett 58:1395–1398, 1987) concerns whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in Gheissari et al. (Commun Math Phys 367(3):771–833, 2019) and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191–225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087–1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.

Horizontal Wavenumber Spectra of Vertical Vorticity and Horizontal Divergence of Mesoscale Dynamics in the Mesosphere and Lower Thermosphere Using Multistatic Specular Meteor Radar Observations

AbstractSpecular meteor radars (SMRs) have significantly contributed to the understanding of wind dynamics in the mesosphere and lower thermosphere (MLT). We present a method to estimate horizontal correlations of vertical vorticity (Qzz) and horizontal divergence (P) in the MLT, using line‐of‐sight multistatic SMRs velocities, that consists of three steps. First, we estimate 2D, zonal, and meridional correlation functions of wind fluctuations (with periods less than 4 hr and vertical wavelengths smaller than 4 km) using the wind field correlation function inversion (WCFI) technique. Then, the WCFI's statistical estimates are converted into longitudinal and transverse components. The conversion relation is obtained by considering the rotation about the vertical direction of two velocity vectors, from an east‐north‐up system to a meteor‐pair‐dependent cylindrical system. Finally, following a procedure previously applied in the upper troposphere and lower stratosphere to airborne wind measurements, the longitudinal and transverse spatial correlations are fitted, from which Qzz, P, and their spectra are directly estimated. The method is applied to a special Spread spectrum Interferometric Multistatic meteor radar Observing Network data set, obtained over northern Germany for seven days in November 2018. The results show that in a quasi‐axisymmetric scenario, P was more than five times larger than Qzz for the horizontal wavelengths range given by ∼50–400 km, indicating a predominance of internal gravity waves over vortical modes of motion as a possible explanation for the MLT mesoscale dynamics during this campaign.

Efficient computation of N-point correlation functions in D dimensions

We present efficient algorithms for computing the N-point correlation functions (NPCFs) of random fields in arbitrary D-dimensional homogeneous and isotropic spaces. Such statistics appear throughout the physical sciences and provide a natural tool to describe stochastic processes. Typically, algorithms for computing the NPCF components have [Formula: see text] complexity (for a dataset containing n particles); their application is thus computationally infeasible unless N is small. By projecting the statistic onto a suitably defined angular basis, we show that the estimators can be written in a separable form, with complexity [Formula: see text] or [Formula: see text] if evaluated using a Fast Fourier Transform on a grid of size [Formula: see text]. Our decomposition is built upon the D-dimensional hyperspherical harmonics; these form a complete basis on the [Formula: see text] sphere and are intrinsically related to angular momentum operators. Concatenation of [Formula: see text] such harmonics gives states of definite combined angular momentum, forming a natural separable basis for the NPCF. As N and D grow, the number of basis components quickly becomes large, providing a practical limitation to this (and all other) approaches: However, the dimensionality is greatly reduced in the presence of symmetries; for example, isotropic correlation functions require only states of zero combined angular momentum. We provide a Julia package implementing our estimators and show how they can be applied to a variety of scenarios within cosmology and fluid dynamics. The efficiency of such estimators will allow higher-order correlators to become a standard tool in the analysis of random fields.

Replicated entanglement negativity for disjoint intervals in the Ising conformal field theory

We calculate the N interval replicated negativity for the Ising conformal field theory using correlation functions of branch point twist fields. For some subset P ⊂ N, this is a calculation of where n is the replica index. This can be reformulated as a calculation of partition functions over superelliptic Riemann surfaces, and for the models in question, this partition function can be expressed in terms of the period matrix of this surface. We detail how to construct the period matrices for these surfaces, giving an analytic expression for . The results are expressed such that when P = Ø, which corresponds to calculating the Rényi entropy, the formulas aligns with known results for the N interval Rényi entropy.

Classical versus quantum intensity-field correlations of scattered light from extended cold atomic clouds

We calculate the intensity-field correlations in the light scattered by $N$ cold atoms driven by a quasiresonant laser field. Fundamental differences occur if the atomic state is an entangled single-excitation state or a coherent factorized state. We provide analytic expressions for the two-time field and intensity correlation functions for the timed Dicke state and the quasi-Bloch state. The comparison with multiatom simulations shows good agreement between numerical and analytic solutions.

Modeling of Multimodal Scattering by Conducting Bodies in Quantum Optics: The Method of Characteristic Modes

We propose a numerical technique for modeling quantum multimodal light scattering by a perfectly conducting body. Using the special quantization technique, we develop the quantum adaptation of the characteristic mode approach widely used in classical electrodynamics. The method is universal with respect to the body's configuration, as well as its dimensions relative to the wavelength. Using this method and calculating the first- and second-order field correlation functions, we demonstrate how scattering affects quantum-statistical features of the field. As an example, we consider the scattering of two single-photon incident Gaussian beams on a cylinder with a circular cross section. We show that the scattering is accompanied by two-photon interference and demonstrates the Hong-Ou-Mandel effect. It is shown that the scattered two-photon field and its correlations manifest directive propagation, which is controllable by various means (angles of incidence, configuration of the body, relations between its dimensions and the wavelength). We expect that this method will be useful for designing quantum-optical devices.

Effectiveness of Random Field Approach in Serviceability Limit State Analysis of Strip Foundation

This work conducts a probabilistic inquiry on how the variability of the parameter defining soil deformability affects the settlement of the foundation located on the soil. The analysis addresses the random foundation model to relevantly estimate the probability of allowable deflection exceedance. The constitutive model parameter is based either on a single random variable or a random field. The computations incorporate direct Monte Carlo sampling and the variance reduction techniques, i.e. Stratified Sampling and Latin Hypercube Sampling. The work focuses on soil parameter modelling by random fields defined by various correlation functions. One of the analytical means is the application of a non-homogeneous function capable of reflecting the soil strata. The probabilistic methods proposed in the paper are tested on a standard example of a foundation strip featuring load eccentricity. It is proved that the modelling mode of the soil—a single variable or a random field—substantially affects the results, i.e. foundation settlements. It was detected that incorporating random fields in soil analysis allows for a valid reliability assessment of a foundation in respect to Serviceability Limit State. The relevantly adjusted correlation functions of random fields allow for a realistic subsoil analysis even in the case of a limited in situ measurement database.

Quantum kinetic theory for spin transport of quarks with background chromo-electromagnetic fields

We derive the quantum kinetic equations for massive and massless quarks coupled with the background chromo-electromagnetic fields from the Wigner-function approach with the n expansion and effective power-counting scheme. For each case, one obtains coupled color-singlet and color-octet kinetic equations, which also involve the scalar and axial-vector components for the charge and spin transport. These kinetic equations delineate entangled evolution of the corresponding distribution functions decomposed in color space. At weak coupling, we derive the close form of the color-singlet kinetic equations for spin transport, which incorporates the diffusion term and the source term that triggers dynamical spin polarization led by correlation functions of color fields. Also, the non-dynamical source term is found in the axial Wigner function. The induced spin polarization and axial charge currents by these source terms are discussed under physical assumptions for color-field correlators in near-equilibrium quark gluon plasmas. In the constant-field limit, we further obtain non-vanishing axial Ward identities, from which we extract the pseudo-scalar condensate for massive quarks at finite temperature.

Scalar quantum fields in cosmologies with2+1spacetime dimensions

Motivated by the possibility to use Bose-Einstein condensates as quantum simulators for spacetime curvature, we study a massless relativistic scalar quantum field in spatially curved Friedmann-Lema\^itre-Robertson-Walker universes with $d=2+1$ spacetime dimensions. In particular, we investigate particle production caused by a time-dependent background geometry, by means of the spectrum of fluctuations and several two-point field correlation functions. We derive new analytical results for several expansion scenarios.

Frequency spectra of horizontal winds in the mesosphere and lower thermosphere region from multistatic specular meteor radar observations during the SIMONe 2018 campaign

In recent years, multistatic specular meteor radars (SMRs) have been introduced to study the Mesosphere and Lower Thermosphere (MLT) dynamics with increasing spatial and temporal resolution. SMRs, compared to other ground-based observations, have the advantage of continuously measuring the region between 80 and 100 km independent of weather, season, or time of day. In this paper, frequency spectra of MLT horizontal winds are explored through observations from a campaign using the SIMONe (Spread-spectrum Interferometric Multistatic meteor radar Observing Network) approach conducted in northern Germany in 2018 (hereafter SIMONe 2018). The 7-day SIMONe 2018 comprised of fourteen multistatic SMR links and allows us to build a substantial database of specular meteor trail events, collecting more than one hundred thousand detections per day within a geographic area of sim 500 km times 500 km. We have implemented two methods to obtain the frequency spectra of the horizontal wind components: (1) Mean Wind Estimation (MWE) and (2) Wind field Correlation Function Inversion (WCFI), which utilizes the mean and the covariances of the line of sight velocities, respectively. Monte Carlo simulations of a gravity wave spectral model were implemented to validate and compare both methods. The simulation analyses suggest that the WCFI helps us to capture the energy of smaller scale wind fluctuations than those capture with MWE. Characterization of the spectral slope of the horizontal wind at different MLT altitudes has been conducted on the SIMONe 2018, and it provides evidence that gravity waves with periods smaller than 7 h and greater than 2 h dominate with horizontal structures significantly larger than 500 km. In the future, these analyses can be extended to understand the significance of small-scale fluctuations in the MLT, which were not possible with conventional MWE methods.Graphical

Massive scalar field in de Sitter spacetime: a two-loop calculation and a comparison with the stochastic approach

We examine long-wavelength correlation functions of massive scalar fields in de Sitter spacetime. For the theory with a quartic self-interaction, the two-point function is calculated up to two loops. Comparing our results with the Hartree–Fock approximation and with the stochastic approach shows that the former resums only the cactus type diagrams, whereas the latter contains the sunset diagram as well and produces the correct result. We also demonstrate that the long-wavelength expectation value of the commutator of two fields is equal to zero both for spacelike and timelike separated points.

Geometry of spacetime from quantum measurements

We provide a setup by which one can recover the geometry of spacetime from local measurements of quantum particle detectors coupled to a quantum field. Concretely, we show how one can recover the field's correlation function from measurements on the detectors. Then, we are able to recover the invariant spacetime interval from the measurement outcomes, and hence reconstruct a notion of spacetime metric. This suggest that quantum particle detectors are the experimentally accessible devices that could replace the classical 'rulers' and 'clocks' of general relativity.

To the numerical simulation of the correlation picture in Dicke superradiance

The paper discusses some approaches to elucidating the picture of electromagnetic field correlations emerging and developing in the Dicke system of two-level atoms interacting via field. Based on the reduced description method, a set of differential equations for the reduced description parameters including field correlation functions is constructed using different quantum evolution pictures and regarding field-matter interaction as a small parameter. Calculations are based on the electrodynamics of continuous media. Meaningful results are obtained in the second-order in interaction. The material equations are found in the class of generalized functions and the material coefficients are estimated in the dependence of coordinates for applying to the spatial description of correlations. Initial and border conditions are discussed. A simplified form of the set of equations is proposed for the initial stage of the process with the supposition about the fixed orientation of emitters acceptable for a one-dimensional system.

Non-Invasive Blood Flow Speed Measurement Using Optics.

Non-invasive measurement of the arterial blood speed gives important health information such as cardio output and blood supplies to vital organs. The magnitude and change in arterial blood speed are key indicators of the health conditions and development and progression of diseases. We demonstrated a simple technique to directly measure the blood flow speed in main arteries based on the diffused light model. The concept is demonstrated with a phantom that uses intralipid hydrogel to model the biological tissue and an embedded glass tube with flowing human blood to model the blood vessel. The correlation function of the measured photocurrent was used to find the electrical field correlation function via the Siegert relation. We have shown that the characteristic decorrelation rate (i.e., the inverse of the decoherent time) is linearly proportional to the blood speed and independent of the tube diameter. This striking property can be explained by an approximate analytic solution for the diffused light equation in the regime where the convective flow is the dominating factor for decorrelation. As a result, we have demonstrated a non-invasive method of measuring arterial blood speed without any prior knowledge or assumption about the geometric or mechanic properties of the blood vessels.

Vibrational Dynamics of Pd-Ni-P Bulk Metallic Glasses: a Local Pseudopotential Study

In the present paper, Phonon modes and elastic constant of three different concentrations of PdxNi1-xP (Pd64Ni16P20, Pd40Ni40P20 and Pd16Ni64P20) bulk metallic glass are calculated using (1) Hubbard-Beeby (HB) and (2) Takeno-Goda (TG) approach along with our well established local model potential. The Hartree (H), Farid et al (F) and Sarkar et al (S) local field correlation functions (LFCF) are employed to examine the effect of the screening function on the collective dynamics of Pd-Ni-P bulk metallic glasses. Results are also reported for phonon dispersion curve, propagation elastic wave and elastic properties viz: bulk modulus BT, modulus of rigidity G, Poisson’s ratio ξ, Young’s modulus Y, Debye temperature ƟD. However, the calculated elastic constants results agree well with other theoretical and available experimental data.