Autocorrelation function equation solution

Induced dipole autocorrelation functions of three-body systems have not yet been computed from first principles. Such work involves the solution of Schrodinger s equation of three interacting atoms. However, classical and semi-classical methods, especially molecular dynamics calculations, exist which offer some insight into three-body dynamics and interactions. Very useful expressions exist for the three-body spectral moments, with the lowest-order Wigner-Kirkwood quantum corrections which were discussed above. 

The equivalence of the Langevin equation (1.1) to the Fokker-Planck equation (VIII.4.6) for the velocity distribution of our Brownian particle now follows simply by inspection. The solution of (VIII.4.6) was also a Gaussian process, see (VIII.4.10), and its moments (VIII.4.7) and (VIII.4.8) are the same as the present (1.5) and (1.6). Hence the autocorrelation function (1.8) also applies to both, so that both solutions are the same process. Q.E.D. 

Before embarking on this discussion one fact must be established. In many applications the autocorrelation function of L(t) is not really a delta function, but merely sharply peaked with a small rc > 0. Accordingly L(t) is a proper stochastic function, not a singular one. Then (4.5) is a well-defined stochastic differential equation (in the sense of chapter XVI) with a well-defined solution. If one now takes the limit rc —> 0 in this solution, it becomes a solution of the Stratonovich form (4.8) of the Fokker-Planck equation. This theorem has been proved officially, but the result can also be seen as follows. 

An alternative explanation of the observed turbidity in PS/DOP solutions has recently been suggested simultaneously by Helfand and Fredrickson [92] and Onuki [93] and argues that the application of flow actually induces enhanced concentration fluctuations, as derived in section 7.1.7. This approach leads to an explicit prediction of the structure factor, once the constitutive equation for the liquid is selected. Complex, butterfly-shaped scattering patterns are predicted, with the wings of the butterfly oriented parallel to the principal strain axes in the flow. Since the structure factor is the Fourier transform of the autocorrelation function of concentration fluctuations, this suggests that the fluctuations grow along directions perpendicular to these axes. 

The signal Scoh represents a convolution integral of the intensity of the probing pulse oc EP(t — to) 2 with the molecular response the latter is governed by the autocorrelation functions 0Vib and 0r- Numerical solutions of Equations (2)-(5) are readily computed and will be discussed in the context of experimental results. 

The velocity autocorrelation function Z t) and its memory function K t) can then be obtained as a self-consistent solution of the above equations. The diffusion and friction coefficients can be obtained as... 

The solution of the ionic velocity Langevin equation v t) for one dimension, leads to an autocorrelation function... 

The Bloch equations (Eq. 5) can be solved under different conditions. The transient solution yields an expression for 0-22 (0> time-dependent population of the excited singlet state S. It will be discussed in detail in Section 1.2.4.3 in connection with the fluorescence intensity autocorrelation function. Here we are interested in the steady state solution (an = 0-22 = < 33 = di2 = 0) which allows to compute the line-shape and saturation effects. A detailed description of the steady state solution for a three level system can be found in [35]. From those the appropriate equations for the intensity dependence of the excitation linewidth Avfwhm (FWHM full width at half maximum) and the fluorescence emission rate R for a single absorber can be easily derived [10] ... 

An uncomplicated solution technique that can be applied to equation (16) is the method of cumulants. In 1972 Koppel showed that the logarithm of the normalized autocorrelation function was identical to the cumulant generating function for the distribution of decay constantsl. The coefficients of the cumulant expansion can be related to the moments of the F(r) distribution. The Koppel equation can be expressed by... 

Equation 13.43 is similar to Equation 13.40. However, the bias due to the shift in the electrostatic reference appears with the opposite sign because the particle that is removed has a positive charge. We have also added a zero-point motion correction. This is necessary because the DETMD simulation as it is usually applied treats all nuclei as classical particles. The approach adopted in [29] is to subtract the zero-point energy of the acid proton attached to the base in full solution. All other protons are classical particles. This effective acid proton zero-point motion is indicated by A pEncx) in Equation 13.43. A pE x) is estimated from the peaks in the (classical) velocity autocorrelation function of the acid proton as obtained from the DETMD trajectory. 

To derive equation (7) it has been assumed that locally the diffusion model is valid. This implies that the changes in the chemical potential of the solute during the decay of the autocorrelation function are sufficiently small to be neglected. The equation, however, properly accounts for the influence of the changes in /iexc(-) across the membrane, arising from the inhomogeneous, anisotropic nature of the bilayer. 

ABSTRACT - The fluorescence anisotropy decay (FAD) technique is first described, then the different expressions ich have been proposed for the orientation autocorrelation function (OACF) of polymer chains are presented. Typical FAD curves of dilute and concentrated solutions of polystyrene labelled with an anthracene group in the middle of the chain are compared to the various OACF expressions and discussed. In the case of bulk polybutadiene, FAD results obtained either on anthracene labelled chains or on 9,10 dialkylanthracene probes free in the polymer matrix, show that the same type of OACF as for polymer solutions can account for the experimental data. Besides, the temperature dependence of the correlation time of the labelled polybutadiene appears to agree with the WLF equation derived from macroscopic viscoelastic measurements, proving that the segmental motions of about 20 bonds which lead to the FAD of labelled polybutadiene participate in the glass transition processes of this polymer. 

The application of this method requires knowledge of the explicit form of at least one of the two funetions (F(/)F(0)> or y t), in order to find a solution of the equation. Variants of the approaeh, developed up to the present time are based on different ways of modeling of the dissipation term y(t), conneeted with the secondary zone of atoms [ 18-20]. Adelman and Garrison use Debye s model for phonons of the solid and obtain an equation for the dissipation term whieh, ean be solved numerically. Doll and Dion propose y t) as a linear combination of conveniently chosen functions, where the coefficients are determined by numerieal self-consistency. Another possibility is to model the microscopic interactions in the lattice of the solid in order to derive a dissipation term. Tully presents the friction as a white noise or positionally autocorrelated function of a Brownian oscillator, including both oscillation and dissipation terms. 

This set of assumptions on the statistical properties of f(t) determines the statistical properties of the solution v(0 of the stochastic differaitial equation in Equation 1.1, which are summarized saying that v(0 is a Gaussian stationary Markov stochastic process, that is, it is generally not delta-correlated. The specific results that follow from this simple mathanatical model regarding propo ties such as the velocity autocorrelation function, msd, and so on, are reviewed in standard statistical physics textbooks [48]. 

Both equations are useful to obtain well-defined D values in each experiment based on a fitting method. Although we understand that the form in Eq. (33.9) is more general, the numerical data from FCS measurement is not sufficient to obtain the full lineshape of D(t) in Eq. (33.9). Seki et al. obtained an analytical solution of autocorrelation curves for D(L) in a step function [39]. They proved that the solution lineshape is different from that of normal diffusion with a non-linear least square algorithm if the deviation from Eq. (33.17) is too small. Even in this case of moderate anomalous diffusion, the observed value of D changes sensitively,depending on f or I. 

One might suppose that the problem is now to solve this system of 2N coupled equations however, one is actually interested in only two aspects of the solution, which are the auto and the total (auto + mutual) orientational correlation functions Consequently, one keeps the two equations for the autocorrelations and < j l (0)6 j (t)>, but sums the equations for the correlations... 

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