On determining a distribution function known only by its moment and/or moment generating function
Distribution function determination using moment generating function and probability density function
Hill, T. W., Jr.↗
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Spherical Harmonic Analysis of Particle Velocity Distribution Function: Comparison of Moments and Anisotropies using Cluster Data
This paper presents a spherical harmonic analysis of the plasma velocity distribution function using high-angular, energy, and time resolution Cluster data obtained from the PEACE spectrometer instrument to demonstrate how this analysis models the particle distribution function and its moments and anisotropies. The results show that spherical harmonic analysis produced a robust physical representation model of the velocity distribution function, resolving the main features of the measured distributions. From the spherical harmonic analysis, a minimum set of nine spectral coefficients was obtained from which the moment (up to the heat flux), anisotropy, and asymmetry calculations of the velocity distribution function were obtained. The spherical harmonic method provides a potentially effective "compression" technique that can be easily carried out onboard a spacecraft to determine the moments and anisotropies of the particle velocity distribution function for any species. These calculations were implemented using three different approaches, namely, the standard traditional integration, the spherical harmonic (SPH) spectral coefficients integration, and the singular value decomposition (SVD) on the spherical harmonic methods. A comparison among the various methods shows that both SPH and SVD approaches provide remarkable agreement with the standard moment integration method.
Gurgiolo, Chris↗
Approximation method for the kinetic Boltzmann equation
The further development of a method for approximating the Boltzmann equation is considered and a case of pseudo-Maxwellian molecules is treated in detail. A method of approximating the collision frequency is discussed along with a method for approximating the moments of the Boltzmann collision integral. Since the return collisions integral and the collision frequency are expressed through the distribution function moments, use of the proposed methods make it possible to reduce the Boltzmann equation to a series of approximating equations.
Shakhov, Y. M.↗
A critical analysis of the Grad approximation for closing out the magnetohydrodynamic equations for plasmas
Error analysis on moment distribution functions of magnetohydrodynamic plasma equations
Meador, W. E.↗
Analytical-numerical methods of calculations of energy and three-dimensional particle distributions in electromagnetic cascades
Analytical and numerical methods of calculation of the energy and three dimensional EPS characteristics are reported. The angular and lateral functions of electrons in EPS have been obtained by the Landau and small angle approximations A and B and compared with earlier data. A numerical method of solution of cascade equations for the EPS distribution function moments has been constructed. Considering the equilibrium rms angle as an example, errors appearing when approximating the elementary process cross sections by their asymptotic expressions are analyzed.
Ivanenko, I. P.↗
Reactive collision terms for fluid transport theory
A modified Bhatnagar-Gross-Krook (BGK) collision method is used for the derivation of reactive collision terms to allow for complete higher-moment approximations in fluid transport theory. The general reactive collision terms for the 8-, 10- and 13-moment approximation for binary reactions are derived. Then, reactive collision terms for specific chemistry are derived with the assumptions: (1) all reactants and products have 13-moment distribution functions; (2) simple, classical assumptions provide sufficient models for reaction dynamics; and (3) the reaction rate is energy-independent. The derived specific formulas reflect two extremes in reactive collision dynamics: direct and indirect reaction mechanisms. Finally, considerations for correct use of the reactive collision terms are discussed in the context of space plasma environments.
Eccles, J. V.↗
Higher moment equations and the distribution function of the solar-wind plasma.
Study of the higher-moment equations for a collisionless fully ionized plasma. For a collisionless, heat-conducting plasma, the distribution function f is cylindrically symmetric about the direction of the magnetic field. It is shown that under a certain assumption the fourth moments of f can be expressed as simple functions of lower moments. Thus no higher-moment terms appear in the third-moment equations. The two third-moment equations, which are obtained in a simple form, join other lower-moment equations to form a closed set of moment equations. The new equations can be used to study the thermal anisotropy and the heat flux of the solar-wind proton. A special case of the cylindrically symmetric distribution function f is found to resemble the proton distribution function reconstructed from solar-wind data, and this resemblance justifies the assumption needed for decoupling the moment equations.
Whang, Y. C.↗
Approximate calculation of cumulative probability from a moment-generating function.
Cumulative probability distribution of positive random variable from moment generating function, exemplifying exponential and Poisson functions
Helstrom, C. W.↗
Determination of moments of the size distribution function in scattering by polydispersions - A comment
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Fymat, A. L.↗
Simulation of spacecraft charging environments by monoenergetic beams
To simulate the distributed spectra of plasmas in space, the differential current density spectrum of the plasma was divided into a number of energy bands and the beam energy and current were calculated for each band to provide a piecewise reproduction of the distributed spectrum. Beam energies and current densities were chosen to match the velocity moments of the plasma distribution function. The velocity moments are averages related to physical quantities such as particle density, flux, pressure, and energy flux, and have been used extensively to characterize the measured properties of plasmas in space. Combinations of one, two, and three beams were found to match two to six velocity moments of Maxwellian distributions. A computational model was used to compare the charging of a spacecraft by plasmas with distributed spectra and by monoenergetic beams.
Halverson, W.↗
A statistical analysis of the low-energy geosynchronous plasma environment. I - Electrons. II - Ions
Data on the geosynchronous plasma environment between approximately 30 eV and 80 keV are analyzed statistically. Nearly 50 days of 10 minute averages of the first four moments of the distribution function from low-energy plasma detectors on the ATS-5 and ATS-6 geosynchronous satellites were used. The data were studied in terms of occurrence frequency, local time variations, and response to geomagnetic activity. These techniques revealed marked differences in the ATS-5 and ATS-6 data bases. When translated into a 2-Maxwellian representation, it was found that (1) the ATS-6 data covered an energy range not covered by ATS-5 (between 1 and 50 eV) and (2) there was a definite change in the ion plasma between the ATS-5 and ATS-6 measurements. Simple expressions are derived to simulate the relationship between the four moments, and a model is presented, which takes into account the ATS-5 and ATS-6 plasma variations. Despite the differences in the ion data, the accuracy of the four-moment representation was found valid for characterizing the geosynchronous electron population.
Garrett, H. B.↗
The mosaic structure of plasma bulk flows in the Earth's magnetotail
Moments of plasma distributions observed in the magnetotail vary with different time scales. In this paper we attempt to explain the observed variability on intermediate timescales of approximately 10-20 min that result from the simultaneous energization and spatial structuring of solar wind plasma in the distant magnetotail. These processes stimulate the formation of a system of spatially disjointed. highly accelerated filaments (beamlets) in the tail. We use the results from large-scale kinetic modeling of magnetotail formation from a plasma mantle source to calculate moments of ion distribution functions throughout the tail. Statistical restrictions related to the limited number of particles in our system naturally reduce the spatial resolution of our results, but we show that our model is valid on intermediate spatial scales Delta(x) x Delta(z) equal to approximately 1 R(sub E) x 1000 km. For these spatial scales the resulting pattern, which resembles a mosaic, appears to be quite variable. The complexity of the pattern is related to the spatial interference between beamlets accelerated at various locations within the distant tail which mirror in the strong near-Earth magnetic field. Global motion of the magnetotail results in the displacement of spacecraft with respect to this mosaic pattern and can produce variations in all of the moments (especially the x-component of the bulk velocity) on intermediate timescales. The results obtained enable us to view the magnetotail plasma as consisting of two different populations: a tailward-Earthward system of highly accelerated beamlets interfering with each other, and an energized quasithermal population which gradually builds as the Earth is approached. In the near-Earth tail, these populations merge into a hot quasi-isotropic ion population typical of the near-Earth plasma sheet. The transformation of plasma sheet boundary layer (PSBL) beam energy into central plasma sheet (CPS) quasi-thermal energy occurs in the absence of collisions or noise. This paper also clarifies the relationship between the global scale where an MHD description might be appropriate and the lower intermediate scales where MHD fails and large-scale kinetic theory should be used.
Ashour-Abdalla, M.↗
Solar wind proton velocity distributions - Comparison of the bi-Maxwellian based 16-moment expansion with observations
The purpose of this paper is to study the possible types of velocity distributions that can be obtained from the bi-Maxwellian based 16-moment expansion of the distribution function, assuming macroscopic parameter values characteristic of the range of solar wind conditions. While previous studies also took heat flow into account, the theoretical expansions for f and the definitions of the physical moments adopted in these studies were different from those used in this paper. The choice of the 16-moment expansion and corresponding moment definitions was motivated by the fact that this is the correct generalization of the widely-used Maxwellian-based 13-moment expansion to the case where the zeroth-order distribution is a bi-Maxwellian. It is found that most of the features characteristic of solar wind proton distributions can be reproduced with the 16-moment distribution, including the appearance of secondary peaks. It is also shown how each of the physically significant velocity moments affects the shape of the distribution function.
Demars, H. G.↗
On the probability of cycle-slipping in first-order phase-locked loops.
The first-passage time boundary value problem for first-order phase-locked loops (PLL) is analyzed, and spectral representations are developed for the probability density function (pdf), the distribution function, and the moments of the first time to passage (or cycle-slip). For the sinusoidal PLL, an asymptotic formula, that is surprisingly accurate even at low loop SNR's and large frequency offsets, is obtained for the pdf of the time to cycle-slip, in terms of the mean time to slip.
La Frieda, J. R.↗
Distribution and moments of radial error
An investigation of the moments and probability distribution of the resultant of two normally distributed random variables is presented. This is the so-called generalized Rayleigh distribution which has many applications in the study of wind shear, random noise, and radar. The most general formula was derived, and two special cases were considered for which tables of the moments and probability distribution functions are included as an appendix. One of the special cases was generalized to n-dimensions.
White, R. G.↗
Plasma and magnetic field characteristics of magnetic flux transfer events
Plasma and magnetic field data from ISEE 1 and 2 are examined for 5 passes of the magnetopause region at 20 and 40 deg northern latitudes, and are presented in terms of moments of the distribution function, calculated from two-dimensional or three-dimensional data. Flux transfer events are characterized by a mixture of magnetosheath and magnetospheric particles, which supports the hypothesis that flux transfer events represent encounters of reconnected flux tubes. An excess pressure appears to be balanced by the tension of the ambient magnetic field lines as they are draped around the reconnected flux tube, and the different observed magnetic field signatures are consistent with expectations for encounters of the flux tubes at different relative locations. It is suggested that increased flow speeds are caused by continued reconnection at the low-latitude boundaries of the flux tubes.
Paschmann, G.↗
Analyses of convective flows and spatial gradients in energetic ion observations
Based on the measured anisotropy of three-dimensional energetic ion distributions obtained from the Medium Energy Particle Instrument experiment of the ISEE 1 spacecraft, bulk flow, epsilon x B convection, and pressure gradients in magnetospheric plasmas are measured with two techniques. In the first technique, using a Compton-Getting transform into the appropriate rest frame in an iterative fashion and converging to the unique velocity that removes the first-order anisotropy from the flux angular distribution, the actual measured angular distribution, and spectrum at each angle, are employed. The second technique uses a first moment of the distribution function in which the shift in velocity space of the distribution function is obtained at two phase space density values. The convection contribution to the velocity shift is assumed to be perpendicular to the pressure gradient contribution if an MHD generator condition is assumed, rendering the two shifts separable, and a gradient scale and a lower limit to the convection electric field are derived in one calculation.
Mitchell, D. G.↗
Numerical investigations of low-density nozzle flow by solving the Boltzmann equation
A two-dimensional finite-difference code to solve the BGK-Boltzmann equation has been developed. The solution procedure consists of three steps: (1) transforming the BGK-Boltzmann equation into two simultaneous partial differential equations by taking moments of the distribution function with respect to the molecular velocity u(sub z), with weighting factors 1 and u(sub z)(sup 2); (2) solving the transformed equations in the physical space based on the time-marching technique and the four-stage Runge-Kutta time integration, for a given discrete-ordinate. The Roe's second-order upwind difference scheme is used to discretize the convective terms and the collision terms are treated as source terms; and (3) using the newly calculated distribution functions at each point in the physical space to calculate the macroscopic flow parameters by the modified Gaussian quadrature formula. Repeating steps 2 and 3, the time-marching procedure stops when the convergent criteria is reached. A low-density nozzle flow field has been calculated by this newly developed code. The BGK Boltzmann solution and experimental data show excellent agreement. It demonstrated that numerical solutions of the BGK-Boltzmann equation are ready to be experimentally validated.
Deng, Zheng-Tao↗