Moment density function

The various studies attempting to increase our understanding of turbulent flows comprise five classes moment methods disregarding probabiUty density functions, approximation of probabiUty density functions using moments, calculation of evolution of probabiUty density functions, perturbation methods beginning with known stmctures, and methods identifying coherent stmctures. For a thorough review of turbulent diffusion flames see References 41—48. 

Moments of a distribution often provide information that can be used to characterize particulate matter. Theyth moment of the population density function n is defined as... 

It can be demonstrated that the total number of crystals, the total length, the total area, and the total volume of crystals, all in a unit of system volume, can be evaluated from the zero, first, second, and third moments of the population density function. 

The magma density Mj- (mass of crystals per unit volume of slurry or Hquor) may be obtained from the third moment of the population density function and is given by... 

Moments of the population density function, which are given by... 

It is clear that an ah initio calculation of the ground state of AF Cr, based on actual experimental data on the magnetic structure, would be at the moment absolutely unfeasible. That is why most calculations are performed for a vector Q = 2ir/a (1,0,0). In this case Cr has a CsCl unit cell. The local magnetic moments at different atoms are equal in magnitude but opposite in direction. Such an approach is used, in particular, in papers [2, 3, 4], in which the electronic structure of Cr is calculated within the framework of spin density functional theory. Our paper [6] is devoted to the study of the influence of relativistic effects on the electronic structure of chromium. The results of calculations demonstrate that the relativistic effects completely change the structure of the Or electron spectrum, which leads to its anisotropy for the directions being identical in the non-relativistic approach. 

The next example will illustrate the technique of calculating moments when the probability density function contains Dirac delta functions. The mean of the Poisson distribution, Eq. (3-29), is given by... 

The physical interpretation of these joint moments is similar in every respect to the interpretation already given for moments of the form ak = E[k]. Thus, a . .. provides a measure of the center of mass of the joint probability density function p 1,...,second order central moments provide a measure of the spread of this density function about its center of mass.30... 

Although we cannot easily obtain expressions for the probability density functions of Y(t), it is a simple matter to calculate its various moments. We shall illustrate this technique by calculating all possible first and second moments of Y(t) i.e., E[7(t)] and E[Y(t)Y(t + r)], — oo < v < oo. The pertinent characteristic function for this task is MYQt (hereafter abbreviated MYt) given by... 

To describe single-point measurements of a random process, we use the first-order probability density function p/(/). Then p/(/) df is the probability that a measurement will return a result between / and / -I- df. We can characterize a random process by its moments. The nth moment is the ensemble average of /", denoted (/"). For example, the mean is given by the first moment of the probability density function. 

Schwerdtfeger, P., Bast, R., Gerry, M.C.L., Jacob, C.R., Jansen, M., Kelld, V., Mudring, A.V., Sadlej, A.J., Saue, T, Sdhnel, T. and Wagner, F.E. (2005) The quadrupole moment of the 3 /2 nuclear groimd state of Au from electric field gradient relativistic coupled cluster and density functional theory of small molecules and the solid slide. Journal of Chemical Physics, 122,124317-1-124317-9. 

Thierfelder, C., Schwerdtfeger, P. and Saue, T. (2007) Cu and Au Nuclear Quadrupole Moments from Four-Component Relativistic Density Functional Calculations using Exact Long-Range Exchange. Physical Review A, 76, 034502-1-034502-4. 

De Proft, F., Martin, J. M. L., Geerlings, P., 1996, On the Performance of Density Functional Methods for Describing Atomic Populations, Dipole Moments and Infrared Intensities , Chem. Phys. Lett., 250, 393. 

A partial differential equation is then developed for the number density of particles in the phase space (analogous to the classical Liouville equation that expresses the conservation of probability in the phase space of a mechanical system) (32>. In other words, if the particle states (i.e. points in the particle phase space) are regarded at any moment as a continuum filling a suitable portion of the phase space, flowing with a velocity field specified by the function u , then one may ask for the density of this fluid streaming through the phase space, i.e. the number density function n(z,t) of particles in the phase space defined as the number of particles in the system at time t with phase coordinates in the range z (dz/2). 

It is well known that the second moment of auniform density function (a function of width x and constant height) is simply x2/12 (Abamowitz, 1965). The standard deviation of this function is then the square root of the second moment that is x/3.464. If the number of samples across the peak is 1, as shown in one of the curves... 

St.-Amant, A., W. D. Cornell, P. Kollman, and T. A. Halgren. 1995. Calculation of Molecular Geometries, Relative Conformation Energies, Dipole Moments, and Molecular Electrostatic Potential Fitted Charges of Small Organic Molecules of Biochemical Interest by Density Functional Theory. J. Comp. Chem. 16, 1483. 

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