Correlation function functional fitting

Santa-Clara and Sornette [67] argue that there are no empirical findings that would lead to a preference of a T-differential or non-differential type of RF. We show that the integrated RF dU t, T) enforces a well-defined short rate process, whereas the non-differential field dW t, T) fails. In the following, we restrict our analysis to these two t5 es of RF models, but keeping in mind that only the T-differential RF ensures a well defined short rate process. Their correlation functions fit with the requirements for a correct modeling of the forward rate curve, while the models remain tractable. 

This form for the correlation function fits a wide variety of relaxation data in complex fluids and, as in the case of many other properties, turns out to have a value close to 0.5. This form for a correlation function is often interpreted in terms of a distribution of relaxation times however in the case of model polyethylene it has been shown that the results cannot be explained in this way. Instead the result is interpreted in terms of a model for anisotropic motion in which the polymer chain is confined to a pipe formed by its neighbors. ... 

Here the vector rj represents the centre of mass position, and D is usually averaged over several time origins to to improve statistics. Values for D can be resolved parallel and perpendicular to the director to give two components (D//, Dj ), and actual values are summarised for a range of studies in Table 3 of [45]. Most studies have found diffusion coefficients in the 10 m s range with the ratio D///Dj between 1.59 and 3.73 for calamitic liquid crystals. Yakovenko and co-workers have carried out a detailed study of the reorientational motion in the molecule PCH5 [101]. Their results show that conformational molecular flexibility plays an important role in the dynamics of the molecule. They also show that cage models can be used to fit the reorientational correlation functions of the molecule. 

FIG. 4 Time-resolved fluorescence Stokes shift of coumarin 343 in Aerosol OT reverse micelles, (a) normalized time-correlation functions, C i) = v(t) — v(oo)/v(0) — v(oo), and (b) unnormalized time-correlation functions, S i) = v i) — v(oo), showing the magnitude of the overall Stokes shift in addition to the dynamic response, wq = 1.1 ( ), 5 ( ), 7.5 ( ), 15 ( ), and 40 (O) and for bulk aqueous Na solution (A)- Points are data and lines that are multiexponential fits to the data. (Reprinted from Ref 38 with permission from the American Chemical Society.)... 

The corresponding gradient-corrected correlation functionals have even more complicated analytical forms and cannot be understood by simple physically motivated reasonings. We therefore refrain from giving their explicit expressions and limit ourselves to a more qualitative discussion of the most popular functionals. Among the most widely used choices is the correlation counterpart of the 1986 Perdew exchange functional, usually termed P or P86. This functional employs an empirical parameter, which was fitted to the... 

The idea of modifying existing functionals by fitting particular terms to accurate experimental data has been tempting to others, too. Stewart and Gill, 1995, have reparameterized a simplified LYP correlation functional formalism, and tested its performance for atomiza-... 

Fig. 4. Neutron spin echo spectra for the self-(above) and pair-(below) correlation functions obtained from PDMS melts at 100 °C. The data are scaled to the Rouse variable. The symbols refer to the same Q-values in both parts of the figure. The solid lines represent the results of a fit with the respective dynamic structure factors. (Reprinted with permission from [41]. Copyright 1989 The American Physical Society, Maryland)...

Though the functional form of the dynamic structure factor is more complicated than that for the self-correlation function, the data again collapse on a common master curve which is described very well by Eq. (28). Obviously, this structure factor originally calculated by de Gennes, describes the neutron data well (the only parameter fit is W/4 = 3kBT/2/C) [41, 44],... 

Fig. 6. A typical correlation function obtained for IFABP in 20 mM phosphate buffer at pH 7.3 at room temperature. The experiment was performed using a ConfoCor 2 LSM combination instrument (Carl Zeiss-Evotec, Jena, Germany) and the correlation function data (G(t)) were fitted to the form G(r) = G(0)/(1 + x/rD), where Td is the characteristic diffusion time. An additional exponential component improves the fit. 

Figure 16 Second Legendre polynomial of the CFI vector autocorrelation function for the sp3 cis-carbon (dashed lines) and the sp2 carbon in a trans-group next to a transgroup (dashed-dotted lines) for two different temperatures. The fit curves to the cis-correlation functions are a superposition of exponential and stretched exponential discussed in the text.

Fig. 53. Fit to the pair correlation function goo for the model consisting of a local pentamer plus continuum (from Ref. 5>).-------neutron diifraction data,-------model calculation...

Most FPA studies to date on DNA have lacked sufficient time resolution to observe directly the relaxation of the internal correlation functions. Instead, the initial anisotropy r0 is taken as an adjustable parameter. Equations (4.30) show that such a procedure is completely valid for anisotropic diffusors (i.e., (A ft)2 = (dx(t)2)), provided the rapid internal motion of the transition dipole is isotropic. It has not yet been ascertained whether the internal motion actually is isotropic, so this must be assumed.(83) A recent claim(86) that large amplitudes of polar wobble are required to fit both the small amplitude of initial FPA relaxation 87 and the linear dichroism88 has been refuted. 83j... 

Fig. 1. Auto-correlation function of the energy and the best fit for a double exponential decay used to obtain the interval of statistical correlation.

Recall that includes the solvation Gibbs energy of the proton, which is not known. Actually, is not needed to calculate the correlation function g j,. However, to compute kj and X2 one needs a value for Xq. Therefore, Xq together with D. and Dj. were fitted to obtain the values of Xj and hence of gj j of succinic acid. Once... 

Fig. 4.14 Results on fully protonated PIB by means of NSE [147]. a Time evolution of the self-correlation function at the Q-values indicated and 390 K. Lines are the resulting KWW fit curves (Eq. 4.9). b Momentum transfer dependence of the characteristic time of the KWW functions describing Sseif(Q,t) at 335 K (circles), 365 K (squares) and 390 K (triangles). In the scaling representation (lower part) the 335 K and 390 K data have been shifted to the reference temperature 365 K applying a shift factor corresponding to an activation energy of 0.43 eV. Solid (dotted) lines through the points represent (q-2 power laws. Full...

Fig. 4.36 Scaling representation of NSE data (density correlation function) corresponding to PI at Q=1.92 A [second maximum of S(Q)]. Times have been divided by the KWW time Tpair to obtain a master curve. T=230 (cross), 240 (empty circle), 250 (plus), 264 (empty square), 280 (empty triangle), 320 K (empty diamond). The solid line indicates the fit with the KWW law for 250 K

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