Correlation functions equation

VER in liquid O 2 is far too slow to be studied directly by nonequilibrium simulations. The force-correlation function, equation (C3.5.2), was computed from an equilibrium simulation of rigid O2. The VER rate constant given in equation (C3.5.3) is proportional to the Fourier transfonn of the force-correlation function at the Oj frequency. Fiowever, there are two significant practical difficulties. First, the Fourier transfonn, denoted [Pg.3041]

Statement 1. Provided K(t) — K — const, i.e., neglecting change in time of the correlation functions, equations (8.2.12) and (8.2.13) of the concentration dynamics describe undamped concentration oscillations with the frequencies u < = y/aj3, dependent on the initial conditions. The de-... 

Figure 5 (a) Stimulated photon echo signal of the azide ion (N3 ) in D2O at 2043 cm-1 as a function of r and T for the — kj + k2 + k3 (gray surface) and the +ki — k2 + k3 (white surface) phase matching directions, (b) Representative traces for four selected values of T. The solid lines represent a global fit of all the scans to the model correlation function Equation (20). (From Ref. 41.)... 

The following equations are used in the calculation of the direct correlation function, Equation (22) (86). 

One still has the problem of relating the potential energy of the system U, which appears in the configurational integral (equation (2.2.32)) and in the pair correlation function (equations (2.4.4) and (2.4.7)) to the intermolecular potential energy between any two molecules, Uy. The potential energy can be expanded in terms of... 

Now making use of the relationship between the average potential energy (W) and the pair correlation function (equation 2.8.8), one obtains... 

The transformation to obtain the difference correlation function equation proceeds in the same way. The difference correlation function matrix is defined by... 

Using the definition of the van Hove self-correlation function (equation (20)) in equation (36) gives ... 

Certain features of the self-correlation triangle centered at the origin are of interest and can be interpreted directly in terms of the structural parameters. In view of the definition of the correlation function, Equation (5.159), we find the following ... 

The Fourier transform method determines the correlation function (Equations 73 and 76) from the scattering data (Stein, 1978). The function F(s) is called a Fourier transform of the structure of scattering elements (particles). 

The correlation function (Equation 48) corresponds to the spectral density (see Equation 26), which represents a superposition of Lorentzians, i.e. of curves like y = alir x +... 

Now let s see how (u(O)u(t)), a property of the fluctuations in equilibrium, is related to a kinetic property, in this case the diffusion coefficient D. Integrate the time correlation function Equation (18.68) over all the possible time lags t. 

Considerations on a more general basis show that Eq. (A.59) not only holds for one-component systems but is generally valid in X-ray scattering experiments. It describes E as the Fourier-transform of the space dependent electron density correlation function. Equation (A.59) is fully equivalent to the Eqs. (A.13), (A.17) and (A.21), when used together with Eq. (A.6). 

In the absence of noise or other absorbing components in the sample set, the value of Cac(d) at = 0 depends on only the absorbance and concentration of the desired component, and all values of CaAd) are zero when d does not equal zero. Under these circumstances, repeated application of the cross-correlation function equation at the various IR frequencies will extract the desired spectrum. 

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