Higher order moments

In strong electric fields contributions to the induced dipole moment that are proportional to or E can also be important, and higher-order moments such as quadrupoles can also be induced. We will not be concerned with such contributions. 

The interpretation of the higher-order moments an is simplified if they are first centered about the first moment. To this end, we define the wth central moment pn of the distribution function or, equivalently,... 

It is often useful when looking at higher order moments first to subtract the mean. This gives us the central moments, the nth central moment is... 

It can readily be seen from this example that the contributions of the extrapolated areas to the total areas are relatively more important for the higher order moments. In this example, the contributions are 28, 61 and 72% for AUC, AUMC and AUSC, respectively. Because of this effect, the applicability of the statistical moment theory is somewhat limited by the precision with which plasma concentrations can be observed. The method also requires a careful design of the sampling process, such that both the peak and the downslope of the curve are sufficiently covered. 

The alternative to compartmental analysis is statistical moment analysis. We have already indicated that the results of this approach strongly depend on the accuracy of the measurement process, especially for the estimation of the higher order moments. In view of the limitations of both methods, compartmental and statistical, it is recommended that both approaches be applied in parallel, whenever possible. Each method may contribute information that is not provided by the other. The result of compartmental analysis may fit closely to the data using a model that is inadequate [12]. Statistical moment theory may provide a model which is closer to reality, although being less accurate. The latter point has been made in paradigmatic form by Thom [13] and is represented in Fig. 39.16. 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

In Chapter 5, we will review models referred to as moment methods, which attempt to close the chemical source term by expressing the unclosed higher-order moments in terms of lower-order moments. However, in general, such models are of limited applicability. On the other hand, transported PDF methods (discussed in Chapter 6) treat the chemical source term exactly. 

The closure problem thus reduces to finding general methods for modeling higher-order moments of the composition PDF that are valid over a wide range of chemical time scales. 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

A two-term power series expression can be derived to handle this case, but it will again fail in cases where the skewness depends on x, but the mean and variance are constant. However, note that the beta PDF can be successfully handled with the two-term form since all higher-order moments depend on the mean and variance. By accounting for the entire shape of the mixture-fraction PDF, (5.318) will be applicable to all forms of the mixture-fraction PDF. 

The agreement between higher-order moments will not be assured. However, this is usually not as important as agreement with the mixture-fraction variance, which controls the rate of micromixing relative to chemical reactions. A complete treatment of this problem is given in Appendix B. 

The term measurable (Billingsley 1979) is meant to exclude poorly behaved functions for which the probability cannot be defined. In addition, we will need to exclude functions which blow up so quickly at infinity that higher-order moments are undefined. 

In the LSR model, the covariances Wu.4> pYn arc known, but not the higher-order moments. Thus, only for first-order reactions will (6.290) be closed ... 

As an example, consider the case of a uniform mean scalar gradient with pi = p2 = 0.5, r = 0 and cT = [ 1 1 ]T (i.e., Ne = 2). For this case, the scalar variance will grow with time due to the lack of scalar variance dissipation (i.e., r = 0 implies that = 0). Moreover, the higher-order moments (up to 2NC — 1 = 3) should approach the Gaussian values.4 The DQMOM representation for this case yields5... 

As noted earlier, it is possible to use JVe < Ns by choosing a linearly independent set of higher-order moments. For example, the multi-environment models discussed in Section 5.10 use jVe = 2-4. 

Data belonging to distribution profiles may be compared either vertically along the release/response ordinate or horizontally along the time abscissa. The semi-invariants (moments) provide a complete set of metrics, representing both aspects in logical sequence AUC accounts (vertically) for the difference of the extent, the mean compares (horizontally) the rates, and higher-order moments and higher-order statistics (variance, etc.) characterize the shape aspect from coarse to finer. 

There are other types of molecular weights based on higher-order moments to the distribution (see Table 1.23), but M and M, are the most widely nsed qnantities. In... 

Note that Equation (9) implies that the square of the standard deviation a2 is the second moment of d relative to the mean d. Higher order moments can be used to represent additional information about the shape of a distribution. For example, the third moment is a measure of the skewness or lopsidedness of a distribution. It equals zero for symmetrical distributions and is positive or negative, depending on whether a distribution contains a higher proportion of particles larger or smaller, respectively, than the mean. The fourth moment (called kurtosis) purportedly measures peakedness, but this quantity is of questionable value. 

Spectral moments can also be computed from classical expressions with Wigner-Kirkwood quantum corrections [177, 189, 317] of the order lV(H2). For the quadrupole-induced 0223 and 2023 components of H2-H2, at the temperature of 40 K, such results differ from the exact zeroth, first and second moments by -10%, -10%, and +30% respectively. For the leading overlap-induced 0221 and 2021 components, we get similarly +14%, +12%, and -56%. These numbers illustrate the significance of a quantum treatment of the hydrogen pair at low temperatures. At room temperature, the semiclassical and quantum moments of low order differ by a few percent at most. Quantum calculations of higher-order moments differ, however, more strongly from their classical counterparts. 

Because the inequalities among the various mean diameters are usually strengthened when the drop sizes are widely dispersed, the ratio of some higher-order moment to a lower-order moment is often useful as a measure of the dispersion of the drop sizes. For example, the coefficient of variation for the surface-weighted size distribution is a function of the ratio of the weight-weighted mean drop size to the volume-surface mean drop size. The variance of the drop-size distribution may also be expressed in terms of the moments of the unweighted size distribution. 

This approximation will be very useful in the following sections. It should be noted that higher-order moments could have been used to generate higher-order approximations. 

Here higher-order moments were neglected.12,28 For this purpose we give a representation of Eq. (4.40) in terms of atomic eigenfunctions similar to (4.34). Moreover, we neglect the second term of (4.41) and write a symmetrical form, we arrive at... 

Another closure approximation relies on the use of higher order moments, but this is almost a deadlock as for only two reacting components, a 13-equation model is required (3). 

For linear systems, the differential equation for the jth cumulant function is linear and it involves terms up to the jth cumulant. The same procedure will be followed subsequently with other models to obtain analogous differential equations, which will be solved numerically if analytical solutions are not tractable. Historically, numerical methods were used to construct solutions to the master equations, but these solutions have pitfalls that include the need to approximate higher-order moments as a product of lower moments, and convergence issues [383]. What was needed was a general method that would solve this sort of problem, and that came with the stochastic simulation algorithm. 

John, 1. G., G. B. Bacskay, and N. S. Hush (1980). Finite field method calculations. VI. Raman scattering activities, infrared absorption intensities and higher order moments SCF and Cl calculations for the isotopic derivatives of H,0 and SCF calculations for CH4. Chem. Phys. 51, 49-60. 

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