Moment method definition

The other two collision source vectors, and can be evaluated using the definitions in Eqs. (6.104) and (6.106). As mentioned earlier, will be closed in terms of the moments of order two and lower, and their gradients. In contrast, C will not be closed in terms of any finite set of moments. Nevertheless, it can be approximated using quadrature-based moment methods as described in Section 6.5. In the fluid-particle limit d d2), neither CI2 i or C will contribute terms involving spatial gradients of the fluid properties (i.e. buoyancy, lift, etc.) to the fluid-phase momentum equation. As mentioned earlier, such terms result from the model for gapi i-n) and would appear, for example, on using the expression in Eq. (6.81). With the latter, Eq. (6.161) becomes... 

To apply the method of moments, the definitions of the /fth moments of the chain length distributions (CLDs) for live and dead polymer, respectively, are invoked ... 

The most common way to calculate average DPs is to use the well-known mathematical method of statistical moments. By definition, the MWD moment of the ith order is given by p = X] The use of this method decreases the number of equations to solve from almost infinite set of equations for R(l) to several differential equations for p . Therefore, in order to calculate Pn, Pw, and Pz, one has to solve only four equations since Pn = Pi/Po, Pw = P hl h-... 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

This expression includes the curvature corrections to the Gaussian function, which play an important role in the averaging procedure they ensure the smoothed spectrum g(E) to be approximated, through the above definition, by its own truncated Taylor expansion. This smoothing procedure of the one-electron spectrum is an application of the method of moments, also used in other systems [23]. 

Measurements of dipole moments, Kerr constants, and dielectric absorption have been employed (81RCR336) widely to obtain information on the conformational equilibrium in acyl heterocycles. Details on conformer structures and populations depend on the choice of additive scheme, group moments, or polarizability tensor in the case of Kerr constants. Several early conclusions, especially for furan- and thiophene-2-carboxaldehyde, appeared contradictory, owing to the choice of these quantities. A more precise definition of polarizability tensors for several heterocycles and a choice of group moments and additive schemes tested on a large amount of available experimental results and supported by accurate theoretical calculations have led to more confidence in the use of experimental dipole moments and Kerr constants in conformational analysis. A limitation of the method is that the... 

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

In previous work, the authors originally arrived independently at two definitions of a reduced free energy in terms of moment densities (moment free energy for short) [31, 32]. Though based on distinctly different principles, the two approaches led to very similar results. We explain this somewhat surprising fact in the present work at the same time, we describe the two methods in more detail and explore the relationship between them. We also discuss issues related to practical applications and give a number of example results for simple model systems. 

We see that the two most widely used methods of polarization moment definition, namely (D.3) and (D.4), differ both in their transformation properties at turn of the coordinate system (which is not always fully realized), and in normalization. In addition, the definition of the matrix element of the operator j Tq, as assumed in [133], does not permit us to use (D.13) directly for the description of optical coherences. 

---