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We review ray-optical methods of analyzing short-wavelength propagation in random media. The advantages and limitations of ray methods are discussed, and results of the statistical theory of ray segment fluctuations pertinent to ray tracing are summarized. The standard method of Monte Carlo ray tracing is compared to a new method which takes into account recent results on the statistics of ray segment fluctuations.
Nonsteady features of a Whitcomb-type supercritical airfoil flowfield were studied in a series of transonic wind tunnel experiments. Data consisted of mean and fluctuating pressures on the airfoil and in the free stream, velocity fluctuations in the wake region, and airfoil vibrations. Fluctuation data were analyzed statistically for intensity, frequency content, and spatial coherence; variations in these parameters were correlated with changes in the mean airfoil flowfield. Surface pressure fluctuation intensity was influenced primarily by the location and motion of the upper-surface shock, the existence of separation, and (downstream of the shock) the development and upstream propagation of 'shocklets'. Power spectra of pressure fluctuations showed characteristics differences up- and downstream of the shock. Chordwise and spanwise coherences were considerably reduced in the presence of separation. Downwash fluctuations at the edge of the wake were noticeably affected only by the development of trailing-edge flow separation.
The interstellar polarization of starlight for an irregularly fluctuating medium is analyzed statistically. A general formulation is presented for the case in which the propagation distance s is larger than the coherence scale of the fluctuations. One specific result for randomly changing field direction is that the linear polarization saturates at a value which can be much less than unity, in agreement with observations.
CFD methods are employing semi-empirical models used in analysis of hypersonic airbreathing engine flow paths. RANS (or Favre averaged) codes have models for turbulent stresses, mass and energy transport; turbulence chemistry interactions. LES methods have models for subgrid scale turbulence. Models depend upon experimental data for validation. Information on mean flow and statistics of the turbulent fluctuation in flow properties is useful. Data requirements: simple well defined supersonic combustion flows with well-known boundary conditions, time and spatially resolved, good instrument precision, and converged statistics. Approach: dual-pump CARS and planar laser-induced fluorescence imaging of OH radical (PLIF).
For this work, insights from non-equilibrium statistical mechanics, highlighting the role of work and fluctuations at the microscale, are applied toward the development of a fundamental, rigorous and purely atomistic theory of nucleation. Nanoscale fluctuations in order, density and heat influence the local nucleation rate by orders of magnitude, necessitating their inclusion through a modern approach. Coarse-graining over the underlying Hamiltonian dynamics allows derivation of a microscale expression for the nucleation rate in terms of a classical path integral over far from equilibrium trajectories and their associated work. Second law violating states at the microscale, as found from the dynamics of small critical nucleation clusters, contribute exponentially to the observable macroscale nucleation rate.
The use is described of several statistical techniques to characterize structure in the angular distributions of secondary particles from nucleus-nucleus collisions in the energy range 24 to 61 GeV/nucleon. The objective of this work was to determine whether there are correlations between emitted particle intensity and angle that may be used to support the existence of the quark gluon plasma. The techniques include chi-square null hypothesis tests, the method of discrete Fourier transform analysis, and fluctuation analysis. We have also used the method of composite unit vectors to test for azimuthal asymmetry in a data set of 63 JACEE-3 events. Each method is presented in a manner that provides the reader with some practical detail regarding its application. Of those events with relatively high statistics, Fe approaches 0 at 55 GeV/nucleon was found to possess an azimuthal distribution with a highly non-random structure. No evidence of non-statistical fluctuations was found in the pseudo-rapidity distributions of the events studied. It is seen that the most effective application of these methods relies upon the availability of many events or single events that possess very high multiplicities.
Calculations using a statistical model of water vapor fluctuations yield the effect of the dynamic wet troposphere on Very Long Baseline Interferometry (VLBI) measurements. The statistical model arises from two primary assumptions: (1) the spatial structure of refractivity fluctuations can be closely approximated by elementary (Kolmogorov) turbulence theory, and (2) temporal fluctuations are caused by spatial patterns which are moved over a site by the wind. The consequences of these assumptions are outlined for the VLBI delay and delay rate observables. For example, wet troposphere induced rms delays for Deep Space Network (DSN) VLBI at 20-deg elevation are about 3 cm of delay per observation, which is smaller, on the average, than other known error sources in the current DSN VLBI data set. At 20-deg elevation for 200-s time intervals, water vapor induces approximately 1.5 x 10 to the minus 13th power s/s in the Allan standard deviation of interferometric delay, which is a measure of the delay rate observable error. In contrast to the delay error, the delay rate measurement error is dominated by water vapor fluctuations. Water vapor induced VLBI parameter errors and correlations are calculated. For the DSN, baseline length parameter errors due to water vapor fluctuations are in the range of 3 to 5 cm. The above physical assumptions also lead to a method for including the water vapor fluctuations in the parameter estimation procedure, which is used to extract baseline and source information from the VLBI observables.
The Jarzynski equality allows the calculation of free-energy differences using values of work measured from nonequilibrium trajectories. The number of trajectories required to accurately estimate free-energy differences in this way grows sharply with the size of work fluctuations, motivating the search for protocols that perform desired transformations with minimum work. However, protocols of this nature can involve varying temperature, to which the Jarzynski equality does not apply. Here, we derive a variant of the Jarzynski equality that applies to varying-temperature protocols, and show that it can have better convergence properties than the standard version of the equality. We derive this modified equality and the associated fluctuation relation within the framework of Markovian stochastic dynamics, complementing related derivations done within the framework of Hamiltonian dynamics.
Direct numerical simulations (DNS) are used to examine the pressure fluctuations generated by fully developed turbulence in supersonic turbulent boundary layers with an emphasis on both pressure fluctuations at the wall and the acoustic fluctuations radiated into the freestream. The wall and freestream pressure fields are first analyzed for a zero pressure gradient boundary layer with Mach 2.5 and Reynolds number based on momentum thickness of approximately 2835. The single and multi-point statistics reported include the wall pressure fluctuation intensities, frequency spectra, space-time correlations, and convection velocities. Single and multi-point statistics of surface pressure fluctuations show good agreement with measured data and previously published simulations of turbulent boundary layers under similar flow conditions. Spectral analysis shows that the acoustic fluctuations outside the boundary layer region have much lower energy content within the high-frequency region. The space-time correlations reflect the convective nature of the pressure field both at the wall and in the freestream, which is characterized by the downstream propagation of pressure-carrying eddies. Relative to those at the wall, the pressure-carrying eddies associated with the freestream signal are larger and convect at a significantly lower speed. The preliminary DNS results of a Mach 6 boundary layer show that the pressure rms in the freestream region is significantly higher than that of the lower Mach number case.
Site Test Interferometers (STIs) have been deployed at two different locations at the NASA Deep Space Network (DSN) tracking complex in Goldstone, California. An STI measures the difference in path length between a geostationary satellite and two antennas on the Earth, producing a measure of atmospheric phase fluctuations over spatial distances comparable to those between elements of possible microwave phased arrays. The purposes of the Goldstone STIs are to assess the suitability of Goldstone as an array site and to statistically characterize atmospheric induced delay fluctuations for application to future array scenarios.The two STI's are separated by 13 km across the Goldstone complex. Each instrument is composed of two small-diameter antennas and associated electronics separated by approx. 200 meters in a principally east-west configuration. The antennas continuously observe signals emitted by geo-stationary satellites and produce data that contain information on the phase difference between signals received by both antennas. The fluctuation in delay (or path length difference) statistics derived from these data sets can be used to infer power loss for particular array configurations.We report on a comparison of the root mean square (RMS) phase delay statistics estimated over 10-minute intervals between the two Goldstone STIs. We have achieved good statistical agreement between the data acquired from the two STIs, given that each instrument is observing different satellites, at different frequencies, over different baseline lengths, with very different implementations, and are located 13 km apart in widely separated terrain at different geodetic altitudes.
It is essential that mesoscopic simulations of reactive systems reproduce the correct statistical distributions at thermodynamic equilibrium. By considering a compressible fluctuating hydrodynamics (FHD) simulation method of ideal gas mixtures undergoing reversible reactions described by the chemical Langevin equations, we show that thermodynamic consistency in reaction rates and the use of instantaneous temperatures for the evaluation of reaction rates is required for fluctuations for the overall system to be correct. We then formulate the required properties of a thermodynamically consistent reaction (TCR) model. As noted in the literature, while reactions are often discussed in terms of forward and reverse rates, these rates should not be modeled independently because they must be compatible with thermodynamic equilibrium for the system. Using a simple TCR model where each chemical species has constant heat capacity, we derive the explicit condition that the forward and reverse reaction rate constants must satisfy in order for the system to be thermodynamically consistent. We perform equilibrium and non-equilibrium simulations of ideal gas mixtures undergoing a reversible dimerization reaction to measure the fluctuational behavior of the system numerically. We confirm that FHD simulations with the TCR model give the correct static structure factor of equilibrium fluctuations. For the statistically steady simulation of a gas mixture between two isothermal walls with different temperatures, we show using the TCR model that the temperature variance agrees with the corresponding thermodynamic-equilibrium temperature variance in the interior of the system, whereas noticeable deviations are present in regions near walls, where chemistry is far from equilibrium.
The diffusion equation which describes the evolution of the average one particle distribution function for an ensemble of strongly turbulent plasmas is derived. The diffusion tensor is a time integral of the autocorrelation tensor of the fluctuations as observed by particles moving along statistically distributed orbits. These orbits contain the effects of fluctuations and thus differ from those encountered in weak turbulence theory. The plasma trajectory equations are used to relate each to the diffusion tensor itself when the turbulence is electrostatic. The diffusion tensor is explicity evaluated for a strongly turbulent unmagnetized plasma.
Pressure fluctuations in a turbulent channel flow are investigated by analyzing a database obtained from a direct numerical simulation. Detailed statistics associated with the pressure fluctuations are presented. Characteristics associated with the rapid (linear) and slow (nonlinear) pressure are discussed. It is found that the slow pressure fluctuations are larger than the rapid pressure fluctuations throughout the channel except very near the wall, where they are about the same magnitude. This is contrary to the common belief that the nonlinear source terms are negligible compared to the linear source terms. Probability density distributions, power spectra, and two-point correlations are examined to reveal the characteristics of the pressure fluctuations. The global dependence of the pressure fluctuations and pressure-strain correlations are also examined by evaluating the integral associated with Green's function representations of them. In the wall region where the pressure-strain terms are large, most contributions to the pressure-strain terms are from the wall region (i.e., local), whereas away from the wall where the pressure-strain terms are small, contributions are global. Structures of instantaneous pressure and pressure gradients at the wall and the corresponding vorticity field are examined.
Pressure fluctuations in a turbulent channel flow are investigated by analyzing a database obtained from a direct numerical simulation. Detailed statistics associated with the pressure fluctuations are presented. Characteristics associated with the rapid (linear) and slow (nonlinear) pressure are discussed. It is found that the slow pressure fluctuations are larger than the rapid pressure fluctuations throughout the channel except very near the wall, where they are about the same magnitude. This is contrary to the common belief that the nonlinear source terms are negligible compared to the linear source terms. Probability density distributions, power spectra, and two-point correlations are examined to reveal the characteristics of the pressure fluctuations. The global dependence of the pressure fluctuations and pressure-strain correlations are also examined by evaluating the integral associated with Green's function representations of them. In the wall region where the pressure-strain terms are large, most contributions to the pressure-strain terms are from the wall region (i.e., local), whereas away from the wall where the pressure-strain terms are small, contributions are global. Structures of instantaneous pressure and pressure gradients at the wall and the corresponding vorticity field are examined.
Equation of state with the quantum statistics corrections is used for particle-number fluctuations ω of the isotopically symmetric nuclear matter with interparticle van der Waals and Skyrme local density interactions. Here, the fluctuations, ω ∝ 1/K, are analytically derived through the isothermal incompressibility K at first order over a small quantum-statistics parameter. Our approximate analytical results appear to be in good agreement with the results of accurate numerical calculations. These results are also close to those obtained by using more accurate Tolman and Rowlinson expansions of the incompressibility K near the critical point. More general formula for fluctuations ω, improved at the critical point, was obtained for a finite particle-number average $\langle$N$\rangle$ by neglecting, for simplicity, small quantum statistics effects. It is shown that for a large dimensionless parameter, α ∝ K 2 $\langle$N$\rangle$/K", where K" is the second derivative of the incompressibility K as function of the average particle density n, far from the critical point (α >> 1), one finds the traditional asymptote, ω ∝ 1/K, for the fluctuations ω. For a small parameter, α << 1, near the critical point, where K = 0 and α = 0, one obtains another asymptote of ω. These fluctuations, having a maximum near the critical point as function of the average density n, for finite values of $\langle$N$\rangle$ are finite and relatively small, in contrast to the results of the traditional calculations.
Scintillation produces fluctuation of the complex envelope of a modulated signal. A useful way to characterize scintillation effects is to describe the signal statistics that result when a CW wave is transmitted through a random medium. Many theoretical treatments describe the signal statistics in a manner identical with the noise theory of Rice. These theories, however, predict Rice statistics only at a very great distance from the perturbing medium, and it has been suspected that generalization to permit the quadrature components of the scattered signal to be partially correlated Gaussian variates might better match the observed signal statistics. Recent tests of signals observed through three types of structured plasma have consistently confirmed this speculation and have revealed a surprising consistency in parameters describing the first-order signal statistics.
CoH3 (CoH ver.3) is an optical model, exciton pre-equilibrium, and Hauser-Feshbach statistical model code, which calculates nuclear reaction cross sections for medium to heavy targets in the keV to MeV energy region. This program is written in standard C++, divided into approximately 200 source and header files. CoH solves the Schroedinger equation for optical potentials defined in the code, and calculates differential elastic scattering, reaction, and total cross sections, for neutron, proton, deuteron, triton, 3He, and alpha-particle. Deformed optical potentials are solved with the coupled-channels method, in which the ground state rotational band members, or vibrational phonon states are coupled. The optical model gives particle transmission coefficients that are fed into the statistical model calculations. CoH includes the pre-equilibrium model (exciton model), the direct/semidirect capture model, and the multi-stage Hauser-Feshbach statistical decay with width fluctuation correction based on the Gaussian orthogonal ensemble. For weakly coupled levels, the DWBA (distorted wave Born approximation) method is used to calculate the direct inelastic scattering process to the excited states.
The inherent non-linearity of intensity correlation functions can be used to spatially distinguish identical emitters beyond the diffraction limit, as achieved, for example, in super-resolution optical fluctuation imaging (SOFI). Here, we propose a complementary concept based on spectral correlation functions, termed spectral fluctuation super-resolution (SFSR) imaging. Through theoretical and computational analysis, we show that spatially resolving time-frequency correlation functions in the image plane can improve the imaging resolution by a factor of $\sqrt2$ in most cases and up to twofold for strictly two emitters. This improvement is achieved by quantifying the degree of correlation in spectral fluctuations across the spatial domain. Experimentally, SFSR can be implemented using a combination of interferometry and photon-correlation measurements. The method works for non-blinking emitters and stochastic spectral fluctuations with arbitrary temporal statistics. This suggests its utility in super-resolution microscopy of quantum emitters at low temperatures, where spectral diffusion is often more pronounced than emitter blinking.