Longitudinal autocorrelation function

When l l, the above gives the so-called cross-correlation functions and the associated cross-correlation rates (longitudinal and transverse). Crosscorrelation functions arise from the interference between two relaxation mechanisms (e.g., between the dipole-dipole and the chemical shielding anisotropy interactions, or between the anisotropies of chemical shieldings of two nuclei, etc.).40 When l = 1=2, one has the autocorrelation functions G2m(r) or simply... 

The expression of the transverse current autocorrelation function can also be derived from the linearized hydrodynamic equations. Because it is decoupled from all the longitudinal modes, the derivation is simple and the final expression in wavenumber and Laplace frequency plane can be written as... 

To obtain an approximate expression for the density autocorrelation function, first we consider that the density fluctuation is coupled only to the longitudinal current fluctuation, and its coupling to the temperature fluctuation and other higher-order components are neglected. 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

The calculation of orientational autocorrelation functions from the free rotator Eq. (14) which describes the rotational Brownian motion of a sphere is relatively easy because Sack [19] has shown how the one-sided Fourier transform of the orientational autocorrelation functions (here the longitudinal and transverse autocorrelation functions) may be expressed as continued fractions. The corresponding calculation from Eq. (15) for the three-dimensional rotation in a potential is very difficult because of the nonlinear relation between and p [33] arising from the kinematic equation, Eq. (7). 

According to von Karmaan and Howarth [179], a consequence of isotropy is that Rij can be expressed in terms of two scalar functions f t,x) and g t, x) identified as the longitudinal and transverse autocorrelation functions, respectively. There are two distinct longitudinal length scales, Af and A/, that can be defined from /, and there are also two corresponding transverse... 

Finally Coffey et al defined the longitudinal relaxation time t as the correlation time, that is, the area under the curve of magnetization autocorrelation function. In this case... 

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