Correlation functions Laplace transform

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

Mw = 2.1 x 106g/mol) in water, which is denoted Cw(t) in the original work [44]. The subscript indicates that both the incoming beam and the scattered light are vertically polarized. The correlation function was recorded for a solution with a concentration of c = 0.005 g/L at a scattering vector of q = 8.31 x 106m-1. The inset shows the distribution function of the relaxation times determined by an inverse Laplace transformation. 

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

A similar approach, also based on the Kubo-Tomita theory (103), has been proposed in a series of papers by Sharp and co-workers (109-114), summarized nicely in a recent review (14). Briefly, Sharp also expressed the PRE in terms of a power density function (or spectral density) of the dipolar interaction taken at the nuclear Larmor frequency. The power density was related to the Fourier-Laplace transform of the time correlation functions (14) ... 

Next, we introduce the Laplace transform of the density time-correlation function,... 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

The integral in this expression is the Laplace transform, Cu (ico) of the time-correlation function, so that... 

From these relations we see that the width and shift of the power spectrum and consequently the spectroscopic lines are related through the Kronig-Kramers dispersion relations. Exactly the same arguments apply to the Laplace transform of the time-correlation function, H(/co). The real and imaginary parts, C H(co) and C"//(/(0), are related by Kramers-Kronig dispersion relation. 

To find the velocity-correlation function corresponding to this memory substitute Eq. (298) into Eq. (296) and then find the inverse Laplace transform... 

Fig. 2.49 Dynamic structure factor at two temperatures for a nearly symmetric PEP-PDMS diblock (V = 1110) determined using dynamic light scattering in the VV geometry at a fixed wavevector q = 2.5 X 10s cm-1 (Anastasiadis et al. 1993a). The inverse Laplace transform of the correlation function for the 90 °C data is shown in the inset. Three dynamic modes (cluster, heterogeneity and internal) are evident with increasing relaxation times.

The direct correlation function and the Laplace transform of the PYg(R) is given analytically. This result has been inverted by Smith and Henderson [10]. 

Equation (37) reduces to Eq. (35). Lebowitz also obtained an analytic expression for the direct correlation function and the Laplace transforms of the gij(R). This transform has been inverted by Leonard et al. [12]. 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

The main results of this sectionjire the out-of-equilibrium generalized Stokes-Einstein relation (203) between Ax2(z) and p(z), together with the formula (206) linking Teff(ffi) and the quantity, denoted as ( ), involved in the Stokes-Einstein relation. One thus has at hand an efficient way of deducing the effective temperature from the experimental results [12]. Indeed, the present method, which avoids completely the use of correlation functions and makes use only of one-time quantities (via their Laplace transforms), is particularly well-suited to the interpretation of numerical data. 

We notice that the correlation function defined by Eq. (147) is stationary. Thus, it fits the Onsager principle [101], which establishes that the regression to equilibrium of an infinitely aged system is described by the unperturbed correlation function. The authors of Ref. 102 have successfully addressed this issue, using the following arguments. According to an earlier work [96] the GME of infinite age has the same time convoluted structure as Eq. (59), with the memory kernel T(t) replaced by (1>,XJ (f). They proved that the Laplace transform of Too is... 

To convince the reader that Eq. (243) is a proper expression for the memory kernel of infinite age, it is enough to show that this approach to the stationary correlation function 3L(t) yields the same result as the prediction of the renewal arguments, namely, Eq. (147). To prove this important fact, we use the definition of the infinitely aged memory kernel given by Eq. (243), we plug it in Eq. (247) and we compare the resulting expression with the Laplace transform of Eq. (147) so as to assess whether we get the same analytical expression. 

Of course, we have to use the exact expression for jjr, (it), of which Eq. (231) is only an accurate approximation. The exact expression for v sta (u) can be found, for instance, in Ref. 96. Then we plug Eq. (248) into Eq. (249) thus obtaining the Laplace transform of the correlation function of c(t), of age ta. The authors of Ref. 102 prove that the result of this procedure is an exact expression for < > (u), which coincides indeed with the recent exact result of Godreche and Luck [103]. 

For this reason, it is convenient to study the Laplace transform of the correlation function of the fluctuation i (f). The Laplace transform of 0 ... 

This poses a problem By applying to the system of Eq. (2.27) the Laplace transformation technique illustrated in Chapter I for evaluating the correlation function... 

As a matter of fact, from Eq. (2.11) the Laplace transform of the correlation function turns out to be... 

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