Cross-correlation terms

Since turbulent fluctuations not only occur in the velocity (and pressure) field but also in species concentrations and temperature, the convection diffusion equations for heat and species transport under turbulent-flow conditions also comprise cross-correlation terms, obtained by properly averaging products of... 

Moreover, because all the selected properties, except for molar refractivity, contain both positive and negative values, these are treated as different properties and cross-correlation terms are also calculated between positive and negative values of each property. Therefore, 7 autocorrelation terms and 12 cross-correlation terms constitute the final TMACC descriptor vector. 

The cross-correlation term is placed on the right-hand side as it acts as a source of dispersion (as does diffusion). We now derive an expression for C. To do so, subtract Equation 38.7 from Equation 38.5 ... 

It is interesting to note that the differences between the SRSS and CQC methods only show up in those bending moments that have contributions of the first and second modes. These modes have closely spaced frequencies and, hence, the cross correlation term for these two modes is significant. Thus, all moments about the y-axis will be in significant error if estimated using the SRSS method. 

Therefore, we conclude that, depending on the response quantity selected, the cross-correlation terms may or may not be important. 

For polymer chains a complicating feature is the presence of cross-correlation terms due to the angular correlations of dipole vectors along a chain (1,6,8). The dielectric increment Ae is a time-averaged (equilibrium) property of a system and may be expressed in terms of the fluctuations of the macroscopic dipole moment TJ(t) of a sphere of volume V according to the Kirkwood-Frohlich relation (1). 

For flexible chains the cross-correlation terms decrease rapidly as k - k l is increased. Model calculations for polyethers (35,36) show that the cross-correlation terms may be positive or negative, giving a Kirkwood factor greater or less than unity (see refs. 5 and 36). 

For the dynamic situation cross-correlation terms are again Important and in place of equation (5) we now have (8,360... 

For multispin systems in which mixing of nuclear spin states occurs owing to dipolar couplings among deuterons, the relaxation is influenced by crosscorrelation terms between individual quadrupolar interaction tensors in addition to the auto-correlation terms [5.17, 5.30]. It is reasonable to expect that these cross-correlation terms may provide additional information on the motional processes. The static Hamiltonian Hq for two deuterons is... 

In principle, these motional spectral densities are involved in determining the decay of the difference magnetization, while the recovery of the sum magnetization is unaffected by the cross-correlation terms. Thus far, the study of relaxation of quadrupolar order in partially ordered systems is straightforward and reliable only when dipolar coupling among resonant spins is weak. 

Eqs. (14) - (19), clearly state the relations between macroscopic and molecular quantities, measurements of e (co) for a given polymer will not (a) allow a separation of the data into the component auto- and cross-correlation terms, (b) give the mechanisms for the decay of auto- and cross-correlation terms. Additional and complementary evidence is required from other experiments in order to do this. This will be discussed in Section II Db below. 

Ago and Ag are the low and high frequency molecular polarizability anisotropy factors, 6 is the angle between the principal axes of groups i and j. Equation (35) indicates that auto-correlation terms (i = j) and P2-type cross-correlation terms,... 

Thus the mean moment that is not relaxed by local motions (p process) is relaxed by the a process, which is an alternative statement of the conservation rule. Quite recendy. Smith and Boyd [40] used a theory to describe the P process (in vinyl acetate copolymers) that is formally equivalent to that described above. The single dipole theory that leads to Equation (10) has been extended by Williams [12] to include cross-correlation terms in the general expression for r) in Equation (8). 

Complete calculation of the proton 7i, taking into account a realistic shape of the spin density distribution with alternating positive and negative values, would be much more complicated than simple use of Eq. (14), as is commonly done. In particular, cross-correlation terms should be included [54]. The result might be somewhat different from that obtained from the simplified theory. 

The fifth relation can be derived by assuming that cross correlations do not occur, as was stated earlier. This yields an extra relation between C,D and E. Calculations on specific models implicitl / assume a definite relationship between "cross-correlation terms" and the normal density and orientation fluctuation terms. 

A comparison between equations 18 and 15 reveals two differences. In the first place the term containing the density correlation function is replaced by a term cT Dq, which means that the model calculations automatically lead to the conclusion that in this case the correlation function for random orientation correlations and that for random density fluctuation correlations are identical. This seems to be a general result. The second difference is the occurrence of cross correlation terms in the equations for B and D. In the case of model calculations these extra terms are connected with the other functions of the system in the way given above. In the general case the cross correlation terms occur only in the B and D value too, but then there need not be a connection with the other functions of the system. 

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