Autocorrelation function Laplace transform

Experimental considerations Frequently a numerical inverse Laplace transformation according to a regularization algorithm (CONTEST) suggested by Provencher [48,49] is employed to obtain G(T). In practice the determination of the distribution function G(T) is non-trivial, especially in the case of bimodal and M-modal distributions, and needs careful consideration [50]. Figure 10 shows an autocorrelation function for an aqueous polyelectrolyte solution of a low concentration (c = 0.005 g/L) at a scattering vector of q — 8.31 x 106 m-1 [44]. 

By taking the Laplace transform of eqn. (290) and then multiplying throughout by the initial velocity u(0) and finally taking an ensemble average, the Laplace transformed velocity autocorrelation function is [490]... 

Note that K(t) is a memory function. Here (S) and K(S) are the Laplace transforms of < >( ) and K(t), respectively. We can now show that the kernel K(t) is related to the autocorrelation function of the random force according to the equation... 

We may also use the results to check that the expressions satisfy the fluctuation-dissipation theorem. The Laplace transform of the autocorrelation function for the fluctuating forces is... 

We see that a calculation of Ar involves a Laplace transform of the time-dependent friction kernel. This may typically be determined in a molecular dynamics (MD) simulation, where the autocorrelation function of the random force (R(O)R(t)) may be determined, which then allows us to determine (f) using the fluctuation-dissipation theorem in Eq. (11.58). Note that Eq. (11.85) is an implicit equation for Ar that in general must be solved by iteration. In the absence of friction we see from Eq. (11.85)... 

From Eqs. (5.2) and (S.5) we see that the diagonal matrix elements of the resolvent and the Laplace transform of an autocorrelation functions are related by... 

The inverse Fourier-Laplace transformation of l/(z — E(z)) provides the autocorrelation function (survival probability) represented in Figure 1.4. The... 

Complete information about the specimen would be available only by tomographic methods with a stepwise rotation of the sample (see e.g. Schroer, 2006) or using inherent symmetry properties of the sample. Under the assumption of fibre symmetry of the stretched specimen around the tensile axis, from the slices through the squared FT-structure the three-dimensional squared FT-structure in reciprocal space can be reconstructed and hence also the projection of the squared FT-structure in reciprocal space. The Fourier back-transformation of the latter delivers slices through the autocorrelation function of the initial structure. Stribeck pointed out that the chord distribution function (CDF) as Laplace transform of the autocorrelation function can be computed from the scattering intensity l(s) simply by multiplying I(s) by the factor L(s) = prior to the Fourier back-... 

Dielectric relaxation measurements couple to the dynamics of the dipole moment of the sample. The dielectric permittivity is the Fourier-Laplace transform of the dipole moment autocorrelation function. 

Remembering that the autocorrelation functions are normalized to unity at t=0, eq. (2.24) means that the Laplace Transforms obey... 

Thns, in either homodyne or heterodyne mode, the DLS autocorrelation function can yield D at any given q from an exponential fit of the form in Eqnation 5.63. For polydis-perse systems, DLS provides an average over the diffnsion coefficient distribution of the particles. A nnmber of approaches have been developed for analyzing polydisperse systems, including the robust cumulant method (see Section 8.2.2), histograms, and the inverse Laplace transform method (see Section 8.2.3). To turn the coefficient distribntion into the MWD requires a known relationship between D and M, often of the form D=AM y. 

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. 

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...

For polydisperse scatterers, the field autocorrelation function may be expressed as the Laplace transform of a continuous distribution F of decay rates ... 

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