Time-correlation function total

This probability distribution is similar to the probability distribution from (7.28), except that here we average only over configurations that have evolved from configurations in. (/ a time t earlier. By integrating this probability distribution over all order parameter values corresponding to region 23 we obtain the total probability that a system initially in 2 is in 23 at time t later. This is nothing other than the time correlation function C(t) and we can write... 

Figure 18. (a) <(5V(5V(t)>. The solid curve at the bottom is the total time correlation function and the curves labeled 1-3 are the contributions of individual shells 1-3. The curve marked /i is the single-particle dipole time correlation function (Cld(t) shown for comparison. From Ref. 57 with permission, from J. Chem. Phys. 89, 5044 (1988). Copyright 1988, American Physical Society. 

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

Here B is an optical constant, or is the total polarizability of the particle, and n is the number of components in each particle. The indexes i and j refer to components of the same particle. If the assumption of independent particles was not made, then the indexes could refer to components of any two particles, and the autocorrelation expression could not be written as a simple sum of contributions from individual particles. The spatial vector r(r) refers to the center of mass of the particle. R(r). In the case of a nonspherical particle (arbitrary shape), Eq. (I0) would describe the coupled motion of the center of mass and the relative arrangement of the components of the particle. For spherical particles, translational and rotational motion arc uncoupled and we have a simplified expression for the electric field time correlation function ... 

To implement the linearized path integral formulation for time correlation functions the initial density operator must be Wigner transformed in the bath variables while it remains an operator in the quantum subsystem space. In the calculations presented below we assume that the system and bath do not interact initially. Consequently total probability density at t = 0 is of the form... 

Time dependence of the normalized total moment-moment time correlation function [ J M(t)] for the water molecules, both in the CsPFO micellar solution (solid line) and in neat water. Circles are the simulation data and the continuous line is a multiexponential fit. 

In the previous sections, we derived general correlation function expressions for the nonlinear response function that allow us to calculate any 4WM process. The final results were recast as a product of Liouville space operators [Eqs. (49) and (53)], or in terms of the four-time correlation function of the dipole operator [Eq. (57)]. We then developed the factorization approximation [Eqs. (60) and (63)], which simplifies these expressions considerably. In this section, we shall consider the problem of spontaneous Raman and fluorescence spectroscopy. General formal expressions analogous to those obtained for 4WM will be derived. This will enable us to treat both experiments in a similar fashion and compare their information content. We shall start with the ordinary absorption lineshape. Consider our system interacting with a stationary monochromatic electromagnetic field with frequency w. The total initial density matrix is given by... 

Since directly measured from the total count rate, only two parameters, f(A) and t, must be obtained from the experimental time-correlation function points. This is an advantage over the analog homodyne method in which, in the corresponding case, the measured h(t) is fit to the form ... 

S is the total relative integrated intensity, So the intensity of the pure translational part, Si the first significant term whose time correlation function depends on intramolecular relaxation times and, Sn = S — (So + Si). 

Dynamic light scattering measurements were performed with a Malvern photon correlation system eqxiipped with a krypton ion laser KR 165-11 from Spectra Physics (1 =647.1 nm). The intensity time correlation function (TCP) was recorded by a Malvern autocorrelator. The electric field TCP g,(t) normalized to the base line of the intensity TCP, and its first cumulant F = -Slng (t)/3t at time to were calculated as usual ( )by an on-line computer where 80 cheinnels of a total of 96 chemnels were used for the recording of the TCP, and the leist 12 channels, shifted by 164 seusple times, were used for the detection of the beise line. 

A rather simple experimental teehnique involving measurement of the time-dependent fluorescence Stokes shift (TDFSS) after an initial exeitation has been applied to measure SD in a large number of liquids. TDFSS oceurs due to dipolar solvation of the excited probe and thus gives an estimate of the solvation timeseales. In an important paper, Jimenez et al. reported the results of SD of the exeited state of the dye coumarin 343 (C343) in liquid water [14]. Their result is shown in Figure 3.13. The initial part of the solvent response of water was found to be extremely fast (few tens of femtoseconds) and it constituted more than 60% of the total solvation energy relaxation. The subsequent relaxation was found to occur in the picosecond timescale. The decay of the solvation time correlation function, S t)y was fitted to a function of the following form... 

Figure 3.13. Comparison of solvation time correlation function S t) and C i) for dye C343 in water. The dashed line shows the experimental result (labeled as expt). The MD simulation result is labeled Aq. Also shown is a simulation for solvation of a neutral atomic solute with the Lennard-Jones parameters of the water oxygen atom (S°). The experimental data were fitted to Eq. (3.9) (using the constraint that the long-time spectrum matched the steady-state fluorescence spectrum) as a Gaussian component (fi equency 38.5 ps 48% of total amplitude) and a sum of two exponential components 126 (20%) and 880 (35%) fs. Adapted with permission from Nature, 369 (1994), 471. Copyright(1994) Nature Publishing Group.

PN(k, t) is the total number density of the solvent whilst pz(k, t) is essentially the (polarization) charge density due to the charge neutrality of the solvent molecule, qb = —In terms of these new quantities, time-correlation functions of number-number (NN), number-charge (NZ), and charge-charge (ZZ) densities can be defined ... 

The article outlines our current understand of the multiple relaxations observed in crystalline and amorphous solid polymers, as tidied laing dielectric techniques An attempt is made to interpret the relaxations of amorphous polymers in a unified way, independent of the details of chemical structure, by use of the time-correlation function approach to partial and total relaxations. In addition, the recent studies of polymers of medium and high degrees of crystallinity are reviewed. 

Apparently, the long time branch of the correlation functions contributes more than 40% to the total coefficient. In order to compute this slowly converging time integral with sufficient accuracy, systems of more than 32 molecules must be used. Modem MD calculations have revealed that the chair form of the time correlation function for Tf appears to depend particularly on the anisotropy of the molecules involved. Luo Hoheisel (1991) have shown that the long-time behavior of the correlation function... 

The effect of 1-D spin dynamics on the ESR line was investigated in various 1-D paramagnetic systems in the 1970s [20-23]. Basically, if the cutoff frequency is less than the linewidth, (Oc < A co, the divergence of f(cu) is felt on the line, which no longer has the Lorentzian shape derived from motional narrowing. The lineshape F(o>) can be analyzed either directly in the frequency domain or in the time domain by considering the total spin time correlation function, G(/) = (S (t)S (O)), with S = Sx-Sx, which is the Fourier transform of F((o) ... 

Dielectric relaxation is sensitive to the time correlation function of the collective variable P(t) that is the total dipole moment, just as quasielastic light scattering is sensitive to the time correlation function of the collective variable YTj= exp(iq r (t)) that is the spatial Fourier component of the concentration. 

The time correlation function C(t) from (19) is a description of the transition statistics in the equilibrium system described in terms of the microscopic degrees of freedom. To make contact with a macroscopic description, appropriate for an experiment in which many molecules of type A and B are present in the sample, it is useful to consider the time evolution of the concentrations ca(i) = NA t)/V and cb (t) = IVb t)/V defined as the number of molecules per volume V of type A and B, respeclively. We imagine that the concentrations ca(1 ) and CB(t) can be determined experimentally in a time-resolved way. Since molecules can only convert into each other and are not created or destroyed, the total number of molecules N = +... 

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