Long-time decay

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

A surprising prediction of the Hong-Noolandi (1978b) theory is the existence of a critical field Fc above which the long-time decay becomes purely... 

FIGURE 7.10 Comparison of long-time decay kinetics for ion-pair recombination. The authors find r 0 ) decay describes the kinetics over a wider range of time than r05. Here T is normalized time. See text for details. Reproduced from Bratczak et al. (1997), with permission of Elsevier . 

Prompt and delayed ionization is familiar for very energy rich molecules. The special feature of high Rydberg states is the initial state that is optically prepared, a state directly coupled to the continuum on the one hand and to a very dense bound manifold on the other. The dynamical theory necessary to describe such states has been reviewed, with special reference to the extremely long-time decay. It is suggested that this resilience to decay is due... 

As discussed in Section IX, there are several approximate schemes to stitch together the short- and long-time dynamic behavior. While the initial decay is determined by the short-time expansion, the long-time decay is determined... 

Fig. 2.16. Observed decay of short-wavelength and long-wavelength bands of DMABN in n-butanol at 21°C and -40°C.78 At room temperature, after about 1 ns, the excited-state equilibrium is established, and both bands show similar long-time decays. At -40°C, however, the equilibration is nearly stopped.

In Figure 3.53 the long-time decay of the excited state is shown with and without factoring in the charge recombination in the bulk, which is negligible at a low concentration of ions (weak excitation). In the latter case there is a false asymptotic behavior at long times, noticed in Refs. 48 and 51 but known even earlier [190] ... 

This disagreement can be accounted for by noticing that the rate provided by the Mori theory is determined from the value of the area below the curve whereas the AEP is equivalent to fitting with an exponential decay the long-time decay behavior of (see Fig. 2). The exact time revolution of 9 (t) can be obtained by means of the diagonalization of the matrix A given... 

This type of problem has recently become of great interest in the field of physics, being directly related to basic problems such as noise-induced phase transitions and the dependence of the relaxation time of the variable of interest on the intensity of external noise terms. As in Chapter X, the reduced equation of motion itself can be assumed to be rehable only when this relaxation time decreases with increasing noise intensity. Further problems of interest are whether or not the threshold of a noise-induced phase transition is characterized by a slowing dowrf " and what is the analytical form of the long-time decay. ... 

Figure 10 A snapshot of the solvation of xenon in acetonitrile together with an ellipsoidal representation of electric field gradient (EFG). The EFG ellipsoid is on average rhombic, and fluctuates both in form and orientation. The fluctuations in the eigenvalues gives a fast vibrational averaging, and the long time decay in determined by reorientation of the EFG principal axis system.

Because of its spin-forbidden nature, the decay rate of the first excited triplet state is extremely low, and very difficult to quantify accurately. What can be measured is the phosphorescence lifetime. Typical long time decays for a poly(bi-spirofluorene) (PSBF) are shown in Fig. 6. 

To account for the additional proton quenching reaction, the long time decay of the photoacid population, Eq. (12.14), should be corrected and take the form of Eq. (12.15) [50-52]. 

A basic feature of the response of fragile liquids to various perturbations is the pronounced nonexponential relaxation behavior. The relaxation function typically exhibits a two-step feature. The fast relaxation at short times is generally associated with vibrational degrees of freedom. The long-time decay of the relaxation function 4>(t), which is governed by the structural relaxation, can often be described by the stretched exponential or the Kohlrausch-Williams-Watts (KWW) function... 

The long-time decay of the dielectric fluctuations is characterized by an angle-dependent diffusion coefficient that decreases linearly with q2 for values of q small compared with qi and q%. 

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