Correlation function Laplace inversion

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. 

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.

If inertial effects are included, the correlation functions pertaining to longitudinal and transverse motions will still be the product of the correlation functions of the free Brownian motion of a sphere [9] and the solid-state correlation functions (cos 19(0) cosi9(t)),p and so on however, the composite expressions will be much more complicated for an arbitrary inertial parameter a, Eq. (102). The reason is that the orientational correlation functions for the Brownian motion of a sphere may only be expressed exactly [18,19] as the inverse Laplace transform of an infinite continued fraction in the frequency... 

The function (t) can be analysed by the method of cumulants [57] or by inverse Laplace transformation. These methods provide the mean relaxation rate T of the distribution function G(T) (z-average). For the second analysis procedure mentioned above, the FORTRAN program CONTIN is available [97,98]. It is sometimes difficult to avoid the presence of spurious amounts of dust particles or high molecular weight impurities that give small contributions to the long time tail of the experimental correlation functions. With CONTIN it is possible to discriminate these artifacts from the relevant relaxation mode contributing to (t). 

Figure 18.16 Intensity-intensity correlation functions for a symmetric PS-PI block copolymer with = 3.4 X 10 Da in benzene at different weight concentrations, (a) G2 q, t) versus w, (b) the Laplace inversion time decay rate distribution GiV) (c) F versus for the three modes. Source Reprinted with permission from Pan C, Maurer W, Liu Z, Lodge TP, Stepanek P, von Meerwall ED, Watanabe H. Macromolecules 1995 28 1643 [87]. Copyright 1995 American Chemical Society.

Any attempt to explain our result of bond reorientation dynamics on the basis of superposition of rotational diffusion processes encounters a contradiction. If such an explanation was to be valid, the same spectrum of D had to be able to explain the shape of the observed Mi(t) and MaCO functions and the broad nature of the reorientation angle distribution W(6,t) at the same time. The spectrum g(x) of correlation time t or the equivalent spectrum g(D) of the rotational diffusion coefficient D can be evaluated from the correlation functions by means of a numerical procedure such as CONTIN [45]. When the correlation function can be represented by an analytical function, the spectrum can be obtained more conveniently by means of inverse Laplace transformation. In the case of a KWW function, with t and p characterizing the function as given in Eq. (12), g(D) can be calculated by [46]... 

The measured intensity time correlation functions were analyzed by inverse Laplace transformation [126, 127], leading to the relaxation rate distributions shown in Figs. 9 and 11. 

The normalized time correlation function, A t), of scattered-light intensity exhibited, at a glance, the presence of two dominant decaying modes for all the solutions studied. The inverse Laplace transformation, ILT, and the... 

In dynamic LLS, the Laplace inversion of each measured intensity-intensity time correlation function (P q, t) in the self-beating mode can lead to a line-width... 

In dynamic LLS, the Laplace inversion of each measured intensity-intensity time correlation function G Hq,t) in the self-beating mode can lead to a line-width distribution GiF), where q is the scattering vector. For dilute solutions, Tis related to the translational diffusion coefficient D by iriq )g- o,c->-o A so that G(L) can be converted into a transitional diffusion coefficient distribution G D) or... 

A r) can be extracted from the measured intensity correlation function g(2)(t) by a computer program that performs the inverse Laplace transformation of g(2)(f) according to Equation 3.29 ... 

Reptation quantum Monte Carlo (RQMC) [15,16] allows pure sampling to be done directly, albeit in common with DMC, with a bias introduced by the time-step (large, but controllable in DMC e.g. [17]) and the fixed-node approach (small, but not controllable e.g. [18]). Property estimation in this manner is free from population-control bias that plagues calculation of properties in diffusion Monte Carlo (e.g. [19]). Inverse Laplace transforms of the imaginary time correlation functions allow simulation of dynamic structure factors and other properties of physical interest. 

Several of the above-described publications extracted rotational spectra from inverse Laplace transforms of imaginary-time autocorrelation functions, quantities readily calculated with RQMC. The utility of defining a larger set of correlation functions, so-called symmetry-adapted imaginary-time autocorrelation functions was explored in a recent paper [50]. Computational efficiency in the calculation of weak spectral features was demonstrated by a study of He-CO binary complex. Some preliminary results of an analysis of a recently observed satellite band in the IR spectrum of CO2 doped He clusters were presented. 

Boutis and coworkers report measurements of the relaxation times of water hydrated N. clavipes and A. aurantia spider silks as a function of temperature by deuterium 2D T1-T2 inverse Laplace transform (ILT) NMR to study the distribution, population and dynamics of hydration water at different temperatures finding correlation times much longer than those for free water and in some cases increasing with increased T. MD simulations reveal that peptides prepared from a number of repeating motifs show inverse temperature transition behavior found for example in protein elastin. 

Fig. 7 Decay of translational (U) and rotational (12) velocity correlations of a suspended sphere. The time-dependent velocities of the sphere are shown as solid symbols the relaxation of the corresponding velocity autocorrelation functions are shown as open symbols (with statistical error bars). A sufficiently large fluid volume was used so that the periodic boundary conditions had no effect on the numerical results for times up to r = 1,000 in lattice units (h = b = 1). The solid lines are theoretical results, obtained by an inverse Laplace transform of the frequency-dependent friction coefficients [175] of a sphere of appropriate size (a = 2.6) and mass (pj/p = 12) the kinematic viscosity of the pure fluid = 1/6...

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