Correlation function multidimensional

For anisotropic scattering patterns and the multidimensional case Vonk ( [168] and [22], p. 302) has proposed to utilize a multidimensional correlation function. It is not frequently applied. 

The equations derived above, describing the A + B —> B reaction kinetics in terms of the correlation functions g and g2, have the form of the nonlinear generalised multi-dimensional diffusion equation. Ignoring the multidimensionality of the operator terms in (5.2.11), these equations could be formally considered as similar to the basic non-linear equations for the A + B — 0 reaction (Section 5.1). Equations studied in both Sections 5.1 and 5.2 are derived with the help of the Kirkwood superposition approximation, the use of which leads to several equations for the correlation functions of similar and dissimilar reactants. 

The present state in the theory of time-dependent processes in liquids is the following. We know which correlation functions determine the results of certain physical measurements. We also know certain general properties of these correlation functions. However, because of the mathematical complexities of the V-body problem, the direct calculation of the fulltime dependence of these functions is, in general, an extremely difficult affair. This is analogous to the theory of equilibrium properties of liquids. That is, in equilibrium statistical mechanics the equilibrium properties of a system can be found if certain multidimensional integrals involving the system s partition function are evaluated. However, the exact evaluation of these integrals is usually extremely difficult especially for liquids. 

Within the quantum formulation of OCT, the basic variational procedure leads to a set of equations for the optimal laser held, which include two Schrodinger equations to describe the dynamics starting from the initial and the target state wavepackets. The optimal laser held is given by the imaginary part of the correlation function for these two wavepackets. This system of equations of optimal control must generally be solved iteratively, making it an extremely computationally expensive approach for multidimensional systems. 

In NMR work, spin-lattice relaxation measurements indicated a non-exponential nature of the ionic relaxation.10,11 While this conclusion is in harmony with results from electrical and mechanical relaxation studies, the latter techniques yielded larger activation energies for the ion dynamics than spin-lattice relaxation analysis. Possible origins of these deviations were discussed in detail.10,193 196 The crucial point of spin-lattice relaxation studies is the choice of an appropriate correlation function of the fluctuating local fields, which in turn reflect ion dynamics. Here, we refrain from further reviewing NMR relaxation studies, but focus on recent applications of multidimensional NMR on solid-ion conductors, where well defined correlation functions can be directly measured. 

In multidimensional NMR studies of organic compounds, 2H, 13C and 31P are suitable probe nuclei.3,4,6 For these nuclei, the time evolution of the spin system is simple due to 7 1 and the strengths of the quadrupolar or chemical shift interactions exceed the dipole-dipole couplings so that single-particle correlation functions can be measured. On the other hand, the situation is less favorable for applications on solid-ion conductors. Here, the nuclei associated with the mobile ions often exhibit I> 1 and, hence, a complicated evolution of the spin system requires elaborate pulse sequences.197 199 Further, strong dipolar interactions often hamper straightforward analysis of the data. Nevertheless, it was shown that 6Li, 7Li and 9Be are useful to characterize ion dynamics in crystalline ion conductors by means of 2D NMR in frequency and time domain.200 204 For example, small translational diffusion coefficients D 1 O-20 m2/s became accessible in 7Li NMR stimulated-echo studies.201... 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

The evolution of the many-molecule dynamics, with more and more units participating in the motion with increasing time, is mirrored directly in colloidal suspensions of particles using confocal microscopy [213]. The correlation function of the dynamically heterogeneous a-relaxation is stretched over more decades of time than the linear exponential Debye relaxation function as a consequence of the intermolecularly cooperative dynamics. Other multidimensional NMR experiments [226] have shown that molecular reorientation in the heterogeneous a-relaxation occurs by relatively small jump angles, conceptually simlar to the primitive relaxation or as found experimentally for the JG relaxation [227]. 

This is the velocity correlation function as obtained by the Wiener-Khinchin theorem. The above arguments may be extended to multidimensional systems, as discussed in some detail by Wang and Uhlenbeck [10] and by McConnell [57] in the context of rotational Brownian motion. 

Consider next the more general case of overlapping resonances. The Q space is now multidimensional and the full effective Hamiltonian matrix of Eq. (4.9) needs to be considered. However, if all the states comprising the Q space are mixing, that is, having vanishing mutual cross correlation functions, than the off-diagonal matrix element of the effective Hamiltonian will also vanish, and Eq. (4.14) is replaced by... 

The possibility of evaluating the coupling coefficients using the analogy of multidimensional infrared (IR) spectroscopy [17] was also considered on the basis of the classical formulation shown in Equation 5.3. In the case of the third-order term aiq q2, the coupling coefficient 0C2 corresponds to the off-diagonal peak intensity of the power spectra, /(Qj, 111, - Here, the power spectra associated with multidimensional IR spectroscopy can be calculated via 2D Fourier transform of the following time correlation function ... 

Perhaps, it was Hynes who initiated two of the most popular so far semi classical non-Markovian approximations [84]. The first approximation was inspired by the success of the [1,0]-Pade approximant, which turns out to be exact in the Markovian limit. This approximation is sometimes referred to as the substitution approximation, because effectively one substitutes non-Markovian two-point distribution function (9.46)-(9.47) into the Markovian expressions (9.50)-(9.51) for the rate kernel. The substitution approximation was shown to work rather well for the case of biexponential relaxation with similar decay times [102]. However, as Bicout and Szabo [142] recently demonstrated, it considerably overestimates the reaction rate when the two relaxation timescales become largely different (see Fig. 9.14). They also showed that for a non-Markovian process with a multiexponential correlation function, which can be mapped onto a multidimensional Markovian process [301], the substitution approximation is equivalent to the well-known Wilemski-Fixman closure approximation [302-304]. A more serious problem arises when we try to deal with the... 

The interpretation of the CDF is straightforward (Stribeclv 2001), since it has been defined by the Laplacian of Vonk s multidimensional correlation function (Vonk, 1979). It presents autocorrelations of surfaces from the scattering entities in that way that positive values characterise distances between surfaces of opposite direction, negative values distances between surfaces of the same direction, see figure 2. 

The MC method consists of generating a set of molecular configurations by random displacements of the N particles in the model and is basically a multidimensional integration procedure that evaluates the integrals for the calculation of the pair correlation function [Eq. (65)] directly. It can be used to calculate the equilibrium properties of a system. 

In summary, we have made an attempt to classify existing MQC strategies in formulations resulting from (i) a partial classical limit, (ii) a connection ansatz, and (iii) a mapping formalism. In this overview, we shall focus on essentially classical formulations that may be relatively easily applied to multidimensional surface-crossing problems. On the other hand, it should be noted that there also exists a number of essentially quantum-mechanical formulations which at some point use classical ideas. A well-known example are formulations that combine quantum-mechanical time-dependent perturbation theory with a classical evaluation of the resulting correlation functions, e.g. Golden Rule type formulations.Furthermore, several... 

If the interaction is not pairwise, similar expressions can be obtained using the corresponding multiparticle correlation functions. If the interaction potential u depends on the direction of the separation vector r as well as on its length r, a multidimensional integration over the components of r is to be performed instead of the spherical averaging leading to integration over r= Irl. 

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