Autocorrelation function, time-dependent

Diffusion. - Distribution of the diffusivitity of fluid in a horizontally oriented cylinder was demonstrated by NMR imaging in two papers on a granular flow system and in the earth s magnetic field. Correlation time (ic) and diffusion coefficient (D = Xc) imaging (CTDCI) was applied to a granular flow system of 2 mm oil-filled sphere rotated in a half-filled horizontal cylinder, ie. to an Omstein-Uhlenbeck process with a velocity autocorrelation function. Time dependent apparent diffusion coefficients are measured, and Tc... 

The experimentally measured intensity autocorrelation function Gjir) depends only on the time interval r, and is independent of t, the time when the measurement... 

FIGURE 9.4. The autocorrelation function of the time-dependent energy gap Q(t) = (e3(t) — 2(0) for the nucleophilic attack step in the catalytic reaction of subtilisin (heavy line) and for the corresponding reference reaction in solution (dotted line). These autocorrelation functions contain the dynamic effects on the rate constant. The similarity of the curves indicates that dynamic effects are not responsible for the large observed change in rate constant. The autocorrelation times, tq, obtained from this figure are 0.05 ps and 0.07ps, respectively, for the reaction in subtilisin and in water. 

I will present here the properties of various sources when the random variable considered is the field intensity. In this case, one has access to the mean and variance via a simple photodetector. The autocorrelation function can be interpreted as the probability of detecting one photon at time t + t when one photon has been detected at time t. The measurement is done using a pair of photodetectors in a start stop arrangement (Kimble et al., 1977). The system is usually considered stationary so that the autocorrelation function, which is denoted depends only on r and is defined by ... 

This expression is derived as the Fourier transform of a time-dependent one-particle autocorrelation function (26) (i.e. propagator), and cast in matrix form G(co) over a suitable molecular orbital (e.g. HF) basis, by means of the related set of one-electron creation (ai" ") and annihilation (aj) operators. In this equation, the sums over m and p run over all the states of the (N-1)- and (N+l)-electron system, l P > and I P " respectively. Eq and e[ represent the energy of the... 

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

Fig. 4. Variation of autocorrelation function with changes in the equilibrium constant in the fast reaction limit. A and B have the same diffusion coefficients but different optical (fluorescence) properties. A difference in the fluorescence of A and B serves to indicate the progress of the isomerization reaction the diffusion coefficients of A and B are the same. The characteristic chemical reaction time is in the range of 10 4-10-5 s, depending on the value of the chemical relaxation rate that for diffusion is 0.025 s. For this calculation parameter values are the same as those for Figure 3 except that DA = Z)B = lO"7 cm2 s-1 and QA = 0.1 and <9B = 1.0. The relation of CB/C0 to the different curves is as in Figure 3. 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

In several papers, only the time-dependent part of the autocorrelation function is considered, and the definition is then... 

Triplet state kinetics can also be studied by FCS (Widengren et al., 1995). In fact, with dyes such as fluoresceins and rhodamines, additional fluctuations in fluorescence are observed when increasing excitation intensities as the molecules enter and leave their triplet states. The time-dependent part of the autocorrelation function is given by... 

The time-dependent quantity in the integrand of Eq. (4.24), (,(r, R, f = 0) ,(r, R, f)), is called the autocorrelation function. It is the integral over all space of the product of the initial wavepacket with the wavepacket at time t. Rama Krishna and Coalson [86] have shown that the Fourier transform over time in Eq. (4.24) can be replaced by twice the half-Fourier transform where the time integral mns from f = 0 to f = oo. Using this result we obtain the final expression ... 

The phase-dependent directionality of photocurrents produced by such a detector entails advantageous properties of the photocurrents cross correlations in nonoverlapping time intervals or spatial regions (considered in Section 4.2.2). These directional time-dependent correlations are measured with one detector only. They involve solely terms dependent on LO phases, in contrast to similar correlations measured by conventional photocounters, which inevitably contain terms depending on photon fluxes such as the LO excess noise. Owing to these properties, the mean autocorrelation function of the SL quadrature is shown in the schemes considered here to be measurable without terms related to the LO noise. LO shot noise, which affects the degree of accuracy to which this autocorrelation is measured (i.e., its variance) is easily obtainable from zero time delay correlations because the LO excess noise is suppressed. The combined measurements of cross correlations and zero time delay correlations yield complete information on the SL in these schemes. 

The final class of dynamical properties we will consider are those defined by time-dependent autocorrelation functions. Such a function is defined by... 

Figure 3.S Two different autocorrelation functions. The solid curve is for a property that shows no significant statistical noise and appears to be well characterized by a single decay time. The dashed curve is quite noisy and, at least initially, shows a slower decay behavior. In the absence of a very long sample, decay times can depend on the total time sampled as well...

In order to be able to use the fluctuation of the intensity around the average value, we need to find a way to represent the fluctuations in a convenient manner. In Section 5.3b in our discussion of Rayleigh scattering applied to solutions, we came across the concept of fluctuations of polarizabilities and concentration of scatterers and the role they play in light scattering experiments. In the present section, what we are interested in is the time dependence of such fluctuations. In general, it is not convenient to deal with detailed records of the fluctuations of a measured quantity as a function of time. Instead, one reduces the details of the fluctuations to what is known as the autocorrelation function C(s,td), as defined below ... 

The theory of spectral moments and line shape is based on time-dependent perturbation theory, Eqs. 2.85 and 2.86, applied to ensembles of atoms, or equivalently on the Heisenberg formalism involving dipole autocorrelation functions, Eq. 2.90. 

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