Moment conditions

The electroneutrality condition can be expressed in temis of the integral of the charge density by recognizing the obvious fact that the total charge around an ion is equal in magnitude and opposite in sign to the charge on the central ion. This leads to the zeroth moment condition... 

The distribution fimctions also satisfy a second moment condition, as first shown by Stillinger and Lovett... 

Mitchell D J, McQuarrie D A, Szabo A and Groeneveld J 1977 On the second-moment condition of Stillinger and Lovett J. Stat. Phys. 17 1977... 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

Dipole a difference in the centers of positive and negative charge in a molecule Dipole Moment condition when the centers of positive and negative charge in a molecule differ... 

As explained in Section VI, we must find functions S.p, Qx which minimize J Eq. (5). What we shall do is solve the modified problem containing the fourth moment condition M, and only at the end of the work allow M to approach its canonical value M. Furthermore, we shall use the formalism of the calculus of variations. 

We note that, in contrast to the simple DH expression, the GDH result for the charge-charge correlation function [276, 278] h r) satisfies the SL second-moment condition (24). When ion pairs are introduced, the SL second-moment condition is, however, only satisfied up to order p2. 

The first and second moment conditions can be very easily introduced into the r5-fit method as least-squares constraints [7,54] if the number of isotopomers is sufficient for a complete restructure. The effect on the coordinates is not expected to be particularly unbalanced unless the moment conditions are required for the sole purpose of locating atoms that could not be substituted (e.g., fluorine or phosphorus) or that have a near-zero coordinate. While all coordinates may change, the small coordinates will, of course, change more. In the cases tested, the coordinate values of the rs-fit with constraints and those of the corresponding r/e-fit (not of the r0-fit), including errors and correlations, differed by only a small fraction of the respective errors, i.e., much less than reported above. This was true under the provision that all atoms could be substituted and that the planar moments that were excluded from the r -fit because of substitution on a principal plane or axis, were also omitted from the r/E-fit. With these modifications, the basic physical considerations and the input data are the same in both cases, and the results should be identical in the limit where the number of observations equals that of the variables. 

Experience has shown the covariance matrix 0rm to be conspicuously different when the same problem was treated by either the rs-method (without enforcing the first and second moment conditions) or any of the r0-derived methods. For the former method the errors of the coordinates were much less correlated. This, as well as the better condition number of the normal equation system, is no doubt a... 

Below a minimal Matlab script for the determination of the CQMOM approximation (Yuan Fox, 2011) for a bivariate case is reported. The script requires the specification of the number of nodes desired for the first (Nl) and second (N2) internal coordinate. The moments used in the calculation must be provided in matrix form. The matrix containing the moments m is defined by two indices the first one indicates the order of the moments with respect to the first internal coordinate (index 1 for moment 0, index 2 for moment order 1, etc.), whereas the second one is for the order of the moments with respect to the second internal coordinate. The structure of the data resembles that of Tables 3.9-3.11. The script calculates the N = N1N2 weights and nodes and stores them in the vector w and in the matrix xi, which have the same structure as in the previous script. The procedure is based on the calculation of the moments with respect to the second internal coordinate, conditioned on the value of the first internal coordinate. The calculation of the quadrature constructed on moments conditioned on the second internal coordinate can be simply carried out by providing the script with the transpose of the moment matrix. 

As in the method, first or second moment conditions cannot be used to evaluate the moments 4- Like the r method, it is therefore restricted to molecules in which all atoms can be substituted. Its advantage compared to the / , method is that it can be used whenever a complete substitution structure is available. This results in a significant reduction of the necessary number of isotopic... 

The b coordinates of all three non-hydrogen atoms were imaginary when the Kraitchman equations were used to locate the position of the nuclei in isothiocyanic acid, HNCS [218]. To determine a reasonable structure for this planar molecule, the authors used the first moment condition for the b coordinates and the product of inertia condition. The second moment condition was used also but with the substitution moment of inertia 4 instead of the ground state moment /q. The authors argued that this procedure gives a more reliable structure instead of a hybrid, particularly since Watson... 

This result was obtained by Chan et al. " and is similar to the second moment condition first derived by Stillinger and Lovett. However, it should be noted that Stillinger and Lovett consider only primitive model electrolytes, and the dielectric constant occurring in their formula is that of the pure solvent. For the mean spherical, LHNC, and QHNC approximations, all three formulas [(5.20b), (5.25), and (5.26)] must give consistently. 

Another valuable quality test for electrolyte solution calculations is based upon the zeroth and second moment conditions of Stillinger and Lovett. To apply these conditions, the first of which is equivalent to the requirement of local electroneutralitywe define the zeroth and second moment defects for an ion of species a as follows ... 

For a single random variable the maximum entropy distribution is obtained by considering only the moment constraints. For the multivariate distribution the correlations between each pair of random variables has to be taken into account as well. This would lead to an optimization problem with 4 optimization parameters with 4 constraints from the marginal moment conditions and ( 4-1)/2 constraints from the correlation conditions, where n is number of random variables. This concept was recently apphed in Soize (2008) to determine the joint density function. For a... 

Accuracy considerations. In the previous section we defined fim as the midpoints of the chosen jic-intervals. If generality is maintained, we are clearly free to make modifications here, and finding that accuracy is thereby improved, we do this. In general, the modified /I s are determined so that one or several moment conditions of the form = l/( + 1) are... 

But each spin is shared by one other tetrahedron, so the 16 configurations per tetrahedron are not independent. Following Pauling s arguments for the water ice problem, 2 = 4 spins are allocated for each tetrahedron and only 6/16 of these satisfy the zero total moment condition. Thus, the true degeneracy, Qq = [2 (6/16)]. Then, the total entropy is 5 = =... 

The distribution functions for electrolytes must satisfy two important moment conditions ... 

Mitchell et al. (1977) showed that the zeroth- and second-moment conditions follow from the asymptotic form of the direct-correlation function [Eq. (33)]. The MS, HNC, RHNC, and Debye Huckel limiting law approximations satisfy both these conditions the Percus-Yevick equation does not obey the second-moment condition and is less useful for electrolytes than it is for uncharged systems. 

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