First-order moments

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

The quantity a is also often referred to as the nth moment of the random variable . We shall use these two designations interchangeably. The first order moment ax (often called the mean of and designated by the symbol m or m ) is a measure of where most of the area under unit quantity of mass distributed on the -axis in such a way that p ( )units of mass are located in a small interval Af centered at the point . 

The limiting case where the chemical time scales are all large compared with the mixing time scale r, i.e., the slow-chemistry limit, can be treated by a simple first-order moment closure. In this limit, micromixing is fast enough that the composition variables can be approximated by their mean values (i.e., the first-order moments (0)). We can then write, for example,... 

For a one-step reaction, most commercial CFD codes provide a simple fix for the fast-chemistry limit. This (first-order moment) method consists of simply slowing down the reaction rate whenever it is faster than the micromixing rate ... 

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... 

In practice, only the first two moments of the random function are of interest. The first order moment is the expectation (mean) of the random function at an arbitrary location x, which is defined to be... 

The zero- and first-order moments give the first-order rotational strength and the second-order rotational strength which are induced for the given transition Aa <- A0 of the achiral molecule ... 

Here / = 1/7 in the standard notation. From our general statements in Section in. A, the spinodal criterion derived from the exact free energy (38) must be identical to this this is shown explicitly in Appendix C. Note that the spinodal condition depends only on the (first-order) moment densities p, and the second-order moment densities py of the distribution p(cr) [given by Eqs. (40) and (41)] it is independent of any other of its properties. This simplification, which has been pointed out by a number of authors [11, 12], is particularly useful for the case of power-law moments (defined by weight functions vt>f(excess free energy only depends on the moments of order 0, 1... K — 1 of the density distribution, the spinodal condition involves only 2K— moments [up to order 2(K — 1)]. 

Hence changes in second- and first-order moment densities are related by... 

We have taken into account here, according to the conventional theory (Pokrovskii 1978), that the relaxation time of the first-order moment is three times bigger than the relaxation time of the moment of the second order in equation (7.49). A solution of equation (7.53) can be written in the form... 

This allows us to understand the reason why the zero- and first-order moments, the oscillator strength and center of gravity of the absorption band,... 

Several possible explanations for the poor A/w predictions can be considered. Since Mw is the problem, a review of the moments equations contained in the base model is in order to see if they are compatible with the way in which is implemented. All of the moment concentration rate equations are based on either the rate of propagation or ratios of termination and chain transfer to the propagation rate. These ratios are very straightforward to correct using the Mayo dimer factor, . The zeroth- and first-order moment rates appear correct, as they are the concentration of terminated chains and the concentration of monomer that has been polymerized, respectively. Mn data are not available to validate the ratio of first to zeroth moment. Calculation of Mw requires the second-order moment, but re-deriving it as a check is beyond the scope of this work. 

In terms of the distribution of relaxation times, the ratio T o /G is an average relaxation time (we may call it "number-average relaxation time"), which is the first order moment of the normalized relaxation spectrum ... 

The moments have physical meaning. The zeroth order moment (iq is the total number of particles per unit volume (or mass, depending on the basis). The first-order moment fii is the total length of the particles per unit volume, with the particles lined up along the characteristic length. The second-order moment is proportional to the total surface area, and the third-order moment is proportional to the total volume. Many physical characteristics of the particles such as the number-mean crystal size, weight-mean crystal size, the variance of the distribution function, and the coefficient of variation also can be represented in terms of the lower order moments of the distribution. 

The centres of a molecule are reference points used to calculate distributional properties of the molecule and, mathematically speaking, are the first-order moments of the considered property. Arithmetic means and weighted arithmetic means are the usual way to calculate centres. 

By weighting atoms by charges, for neutral molecules, the first order moment of the charges is the - dipole moment. Moreover, for molecules with zero net charge and nonvanishing dipole moment, the centre-of-dipole was recently defined as the appropriate centre for multipolar expansions to obtain rotational invariance [Silverman and Platt, 1996]. 

The disperse-phase mean momentum equation is found from the first-order moment of the velocity distribution function ... 

The moment-transport equations that we have derived up to this point are of first order in the velocity variables. In order to describe fluctuations about the first-order moments, it is necessary to derive transport equations for second- and sometimes higher-order moments. Just as before, this is accomplished by starting from Eq. (4.39) with a particular choice for g. In order to illustrate how this is done, we will consider the function g = fp2V which results in the particle-mass-average moment... 

The reader can verify that the source terms for the zeroth- and first-order moments are exactly zero (i.e. the conservation of mass and mean momentum). It is interesting to... 

Table 6.4. v, g) terms for first-order moments with m = 1,2,3... 

Thus, since the terms for the first-order moments in Table 6.4 are the same for g and -g, it follows that Cjqo = oio = nonzero contribution from Eq. (6.71)... 

As with the first-order moments, this expression has contributions due to the kinetic and collisional fluxes on the left-hand side, and due to the collision source terms on the right-hand side. The contributions due to like-particle collisions (Oi,200,11 and C2oo,ii) have the same forms as in the case of monodisperse particles described above. We will thus look briefly at the terms due to unlike-particle collisions. 

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