Correlation function long-time behavior

Long-time behavior of correlation functions. The dipoles induced in successive collisions are correlated as Fig. 3.4 on p. 70 suggests. As a consequence, the dipole autocorrelation function has a negative tail of a duration comparable to the mean time between collisions, Fig. 5.3. Furthermore, the area under the negative tail is of similar order of magnitude as the area under the positive (or intracollisional) part of C(r). If the neg-... 

Computer experiments on condensed media simulate finite systems and moreover use periodic boundary conditions. The effect of these boundary conditions on the spectrum of different correlation functions is difficult to assess. Before the long-time behavior of covariance functions can be studied on a computer, there are a number of fundamental questions of this kind that must be answered. 

The complex nature of the slow mode responsible for the long-time behavior of first rank correlation functions for a first rank interaction potential is illustrated by the composition of the eigenvector corresponding to the slow mode 11a in Table XI, for Uj = 3 and o) = 0.5. Note that n 1, tij, ii, J2 describe the magnitudes and the orientations of the momentum vectors Lj and L2 j is referred to the orientation of L, -t- Lj, 7, and J2 are related to the orientations of the two bodies, and the total orientational angular operator defines the quantum number J finally J, which is not included in this table, is the total angular momentum quantum number, and it is always equal to 1 for first rank orientational and momentum correlation functions, and to 2 for second rank correlation functions. In Fig. 11 we show the first rank correlation functions for different collision frequencies of body 1. The second rank correlation function decays are plotted in Fig. 12. The librational motions in the wells are more important than they were in the first rank potential case (since there is now a more accentuated curvature of the potential wells). 

In theoretical studies, one usually deals with two simple models for the solvent relaxation, namely, the Debye model with the Lorentzian form of the frequency dependence, and the Ohmic model with an exponential cut-off [71, 85, 188, 203]. The Debye model can work well at low frequencies (long times) but it predicts nonanalytic behavior of the time correlation function at time zero. Exponential cut-off function takes care of this problem. Generalized sub- and super-Ohmic models are sometimes considered, characterized by a power dependence on CO (the dependence is linear for the usual Ohmic model) and the same exponential cut-off [203]. All these models admit analytical solutions for the ET rate in the Golden Rule limit [46,48]. One sometimes includes discrete modes or shifted Debye modes to mimic certain properties of the real spectrum [188]. In going beyond the Golden Rule limit, simplified models are considered, such as a frequency-independent (strict Ohmic) bath [71, 85, 203], or a sluggish (adiabatic)... 

Finally, we should mention that there are other approaches to determining the long-time behavior of the time correlation functions, which are based on solutions of the linearized hydrodynamic equations. These approaches also lead to a decay for the correlation function, and for po t)... 

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 persistence implies that long time intervals in which the process does not jump between states on and off, control the asymptotic behavior of the correlation function. The factor P+, which is controlled by the amplitude ratio A+/A-, determines the expected short and long time t behaviors of the correlation function, namely C(oo, 0) = lim oo (7(f)/(f + 0)) = P+ and C(oo,oc) = lim oo(7(f)/(f + oo)) = (P+)2. In slightly more detail the two limiting behaviors are... 

The pressure-drop correlations outlined above assume a constant value of e, the bed void fraction. In industrial-hydroprocessing trickle-bed operations, such as in a hydrodcsulfurization reactor, the pressure drop has been found to increase with time. A typical behavior of the pressure drop across an industrial HDS reactor as a function of time is shown in Fig. 6-4. The pressure drop remains essentially constant over a long initial period, where the correlations given above should be useful. After a while, however, as shown in Fig. 6-4, the pressure drop increases very rapidly with time until the operation requires termination due to an excessive pressure drop across the bed. 

Equations (4.4) and (4.S) are reminiscent of each other in the sense that the diffusional behavior of water in the long-time region results from different, randomly explored, environments, each one being characterized by well-defined diffusion coefficients. Therefore, the corresponding macroscopic correlation function turns out to be an average over several molecular environments, the permanence time in each environment and the transition rate from one to another being determined by a well-defined statistical process. 

Note that the exact form of the above ansatz is not critical for obtaining the general behavior of C(t), as described below, the important parameters being the decay time of the modulation correlation function and the mean square modulation depth . The analytical expression for C(t) is now differentiable to second order and exponential decay is followed only at long times t > Tp. Exjjerimental nonjjerfectly Lorentzian Raman line shapes can often be fitted with this equation. 

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