Normalized correlation functions

The correlation time, given by l/ eff> may be calculated exactly in the following way. Let us define the normalized correlation function of a stationary process by... 

The subtraction of the average (r) guarantees that the normalized correlation function T(f) vanishes for large times. Obviously, T(7j is normalized according to T (0) = 1. A correlation time may be defined by... 

In this Gaussian modulation case, the normalized correlation function of the moment x(t) is given by... 

Fig. 7 The correlation function (tiT2)d as obtained from kinetic Monte Carlo simulations for the polypeptide model (green). The normalized correlations functions (ki 2fc "2) (red) and (R1R2) (black) are also shown for the sake of comparison. All correlation function are normalized so that their initial value is equal to 1. The following parameters were used under different conditions (1) Folded state ko = = 2000,...

The normalized correlation function can be related to the distribution of decay constants as shown by Equation 2. 

Unlike case 1, the correlation function approaches zero when t —> oo this is because if t is large, we expect to find the process in state off. Using Eq. (16), the asymptotic behavior of the normalized correlation function Eq. (11) is... 

With this assumption the normalized correlation function — t2) becomes... 

Figure 6. (a) 2-Gilorobutane at 50 K, 6x6 site-site potential, angular velocity autocorrelation functions. Crosshatdiing indicates computer noise difference between R and S enantiomers. (—) Racemic mixture, (b) As for (a), under the influence of a strong field E, producing a torque — 63 XE in each molecule of the molecular dynamics sample. (1) (—) Racemic mixture (2) (—) R enantiomer. Ordinate Normalized correlation function abscissa time, ps. 

The response of a dipole, surrounded by neighbours, to an applied fidd can in fact be related in detail to the frequency variation of the complex permittivity. A theoretical treatment of the effect on the (fipole of the lag in the reaction field has been given by Scaife and by Fatuzzo and Mason. Equation (14) has been derived by Klug et al. and also by Nee and Zwanzig, in which y(t), defined in equation (IS), is the normalized correlation function of the permanent dipole moment /r. The function... 

It is interesting that the normalized memory function for , which has non-Gaussian structure, lies very dose to the normalized correlation function /< I > for the angular momentum vector, while the memory function for the angular momentum is very neariy Gaussian. The former correspondence is not surprising, since the motion of a linear molecule is equival it to that of a particle on a sphere, so that the Kubo-Shimizu consideration of a plane rotation may apply with only the numerical change which makes the relevant correlation function in the exponent of Kubo s formula. 

Figure 2. log of normalized correlation function C(t) as a function of time for sample 64-2, with c = 1.18 mglmL. 

FIG. 7. Normalized correlation functions obtained from the analysis of the hole width in acetonitrile (1) and methanol (2). Dashed curves are reported Pm (f) of LDS750 in acetonitrile and 1-aminonaphtalene in metlmol, see text. 

Here, as mentioned above, we have used (22) to combine the two Gaussians in (19) in a single Gaussian function evaluated at zero time. The symbol in this equation indicates that some irrelevant constants have been dropped, not that there are approximations in this result. Below we replace it with an equality as the constants which normalize the propagators divide out when the appropriately normalized correlation function is computed. The semi-classical amplitude can be conveniently re-expressed by introducing a polar representation of the complex polynomials identifying the initial and final occupied mapping states, thus... 

Since we are interested in the time-dependence of <<57/(0)<57/(/)> and not in its absolute value, it is convenient to work with the normalized correlation function. In Appendix 6.A we show that this can be written as... 

If we consider a mass fractal object as a distribution of mass points, the normalized correlation function y(r) is the probability of finding a mass point a distance r apart from an arbitrary mass point selected within the object. We construct a spherical shell of radius r and thickness dr around the selected point. The number of mass points enclosed in the shell is proportional to Airrd l dr in view of Equation (5.114). Since the volume of the shell is equal to Artr2 dr, the correlation function is... 

If I q) is known only in arbitrary units, as is usually the case in practice, the normalized correlation function y (x) may instead be evaluated by... 

It is convenient to discuss normalized correlation functions r,y, defined hy the equation... 

Figure 9.14. Survival probability S(t) of an irreversible ET reaction [Pi(t) in our notation] in the solvent-controlled regime (k -> co) for biexponential solvent relaxation model with the normalized correlation function A(t) The parameter AG is the activation... 

The normal-normal correlation function can be used to define the persistence length of the membrane as the distance over which the normal becomes decorrelated via the thermal undulations the distance r at which gnir) is of order unity. The persistence length, is defined as... 

The related theory about extenics is founded by the Chinese scholars Cai wen to solve the problem of subjective and objective contradictions in 1983 (Cai, 1983), it bases on the matter-element theory and extension set theory and does research on the influence degree of the described problems about quantity and quality , so as to completely know the system features (Guo et al., 2009). Due to the complexity production conditions of underground working face, there are a variety of factors that affect gas emission, how to quickly and accurately judge the reasons that why the gas emission is abnormal is an important task of gas early-warning, the extension theory can calculate the abnormal gas emission level by using the normalized correlation function. Under the condition of correct identification of gas hazard in coal mine. 

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