Correlation functions scalar

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

Using the summation theorem for spherical harmonics, these correlation functions may be represented as scalar products... 

It must be stressed that every Un(t,J) brings dq(0) to a different coordinate system. Consequently, the averaged operator (A7.13) is actually a weighted sum of the quantities in differently oriented reference systems. It can nevertheless be used to find the scalar product (A7.7), that is the orientational correlation function. 

To simplify the discussion, a scalar even variable x will be used. In this case the most likely terminal position is x (x, x) = —Q(x)Sx, where the correlation function is <2(x) = 1 (x(t + x)x(f))0. The most likely terminal velocity is... 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

Note that evaluating the correlation functions at r = 0 yields the corresponding one-point statistics. For example, Rap(0, t) is equal to the scalar covariance W,/prp). 

Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend only on r = r, i.e., R,p(r, t). The scalar integral scale L and the scalar Taylor microscale >-,p can then be computed based on the normalized scalar spatial correlation function fp, defined by... 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

For homogeneous scalar fields, the scalar spectrum 4> (k, t) is related to the scalar spatial correlation function defined in (3.39) through the following Fourier transform pair (see Lesieur (1997) for details) ... 

Note that from its definition, the scalar spatial correlation function is related to the scalar variance by... 

Scalar correlation at the diffusive scales can be measured by the scalar-gradient correlation function ... 

Owing to the sensitivity of the chemical source term to the shape of the composition PDF, the application of the second approach to model molecular mixing models in Section 6.6, a successful model for desirable properties. In addition, the Lagrangian correlation functions for each pair of scalars (( (fO fe) ) should agree with available DNS data.130 Some of these requirements (e.g., desirable property (ii)) require models that control the shape of /, and for these reasons the development of stochastic differential equations for micromixing is particularly difficult. 

In particular, for scalars with different Schmidt numbers the Lagrangian correlation function must exhibit the correct dependence on Sc and Re. 

As indicated, the power law approximations to the fS-correlator described above are only valid asymptotically for a —> 0, but corrections to these predictions have been worked out.102,103 More important, however, is the assumption of the idealized MCT that density fluctuations are the only slow variables. This assumption breaks down close to Tc. The MCT has been augmented by coupling to mass currents, which are sometimes termed inclusion of hopping processes, but the extension of the theory to temperatures below Tc or even down to Tg has not yet been successful.101 Also, the theory is often not applied to experimental density fluctuations directly (observed by neutron scattering) but instead to dielectric relaxation or to NMR experiments. These latter techniques probe reorientational motion of anisotropic molecules, whereas the MCT equation describes a scalar quantity. Using MCT results to compare with dielectric or NMR experiments thus forces one to assume a direct coupling of orientational correlations with density fluctuations exists. The different orientational correlation functions and the question to what extent they directly couple to the density fluctuations have been considered in extensions to the standard MCT picture.104-108... 

In subsection 3.1, we will present GGA and LDA calculations for Au clusters with 6first principles method outlined in section 2, which employs the same scalar-relativistic pseudo-potential for LDA and GGA (see Fig 1). These calculations show the crucial relevance of the level of density functional theory (DFT), namely the quality of the exchange-correlation functional, to predict the correct structures of Au clusters. Another, even more critical, example is presented in subsection 3.2, where we show that both approaches, LDA and GGA, predict the cage-like tetrahedral structure of Au2o as having lower energy than amorphous-like isomers, whereas for other Au clusters, namely Auig, Au ... 

The classical dipole correlation function is symmetric in time, C(—t) = C(f), as may be seen from Eq. 5.59 by replacing x by x — t the classical scalar product in Eq. 5.59 is, of course, commutative. Classical line shapes are, therefore, symmetric, J(—. Furthermore, classical dipole autocorrelation functions are real. 

If the symmetry is different, then of course iL /, > can be nonzero. In this article we assume that 0t,..., VN have definite albeit different time reversal symmetries. The properties can be represented by vectors t/j >... t/jy>... in Hilbert space with scalar product defined above. It is a simple matter to demonstrate that L is Hermitian in this Hilbert Space. Define the time correlation function... 

This function describes the correlation between 0,(0) and Oj(t) as a function of the time. Corresponding to each definition of the scalar product (Eq. (73)) there is a different correlation function. 

It is possible to derive an equation which describes the time evolution of the time-correlation function Cn(t) where C stands for different autocorrelation functions depending on the definition of the scalar product (i), (ii), or (iii) of Eq. (73) adopted. 

All-electron DFT calculations were performed using the DMOL [24] code. These incorporated scalar relativistic corrections and employed the non-local exchange and correlation functional Perdew-Wang91 [25] denoted GGA in the rest of the paper, which is generally found to be superior to the local density approximation (EDA)... 

The generalized RISM equation (Equation [77]) then becomes a single scalar equation (essentially an average correlation function)... 

Simultaneous Rotation and Fluctuation. The scalar product p(0)-(r(t) has the value /n(0) /x(/) cos 6(0 so the dipole auto-correlation function is (fdP) MO cos 6(/)>, where 6(t) is the angular displacement of the dipole axis and MO), MO the relevant magnitudes. The correlation function will separate into free rotation. If the variation of dipole moment and molecular size is relatively small, the correlation between that and the rotation may be negligible. In that case, if is independently known, may be obtained from an experimentally determined -... 

The advantage of the correlation function approach is that only the storage of scalar quantities, rather than wave packets, is needed. Thus, the memory requirement is significantly reduced, an issue that may become more important for large systems. The implementation with the Chebyshev propagator takes further advantage of its numerical properties discussed above. In cases where resonances are dominant, the LSFD approach can be used to further reduce computational costs. We note in passing this approach can be extended to the calculation of thermal rate constants. 

To calculate Qu t,x), the v- terms must be determined. Therefore, the scalar function /(r) called the Eulerian correlation function for isotropic turbulence in sect 1.3, is introduced (e.g., [88], chap 8 [7], chap 8 [5], sect 3.4) ... 

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