Autocorrelation function angular velocity

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

In the following we focus our attention on approximate velocity and angular momentum autocorrelation functions generated from postulated memory functions. The theory behind these approximations has been outlined previously in this section. Each of the proposed memory functions that we shall consider has already been discussed in the previous sections. Here we examine how well the time-correlation functions generated from these postulated memories reproduce our experimental correlation functions and spectra. It is also informative to see the relationships between the postulated and experimental memories for our systems. 

Each of these postulated memories was used to solve the appropriate Volterra equation numerically for the approximate autocorrelation functions / (0 and A j(t) (see Appendix B). Three different experimental autocorrelation functions were tested the velocity autocorrelation function from both the Stockmayer and modified Stockmayer simulations and the angular momentum autocorrelation function from the modified Stockmayer simulation. The parameters needed by the postulated memory functions for each of these three autocorrelation functions are tabulated in Table IV. 

Finally, consider the power spectra of the experimental approximate correlation functions which are displayed in Figures 24, 29, and 34. Note that each of these spectra has been normalized to unity at co = 0. Note also that the experimental spectrum from the angular momentum correlation function is much broader than the experimental velocity autocorrelation power spectra. The power spectra of the Gaussian II autocorrelation functions are in much better agreement with the experimental spectra at all frequencies than the power spectra of the other approximate autocorrelation functions. 

The experimental memories for our velocity and angular momentum autocorrelation functions decay initially to approximately zero in a Gaussian fashion. 

The important point to note here is that the 2nd moment of Ky(t) depends on the 2nd and 4th moments of y(t). The 2nd moments of each of the three previously mentioned autocorrelation functions may be calculated from ensemble averages of appropriate functions of the positions, velocities, and accelerations created in the dynamics calculations. Likewise, the 4th moment of the dipolar autocorrelation function may also be calculated in this manner. However the 4th moments of the velocity and angular momentum correlation functions depend on the derivative with respect to time of the force and torque acting on a molecule and, hence, cannot be evaluated directly from the primary dynamics information. Therefore, these moments must be calculated in another manner before Eq. (B.3) may be used. 

Fig. 4. (a) The velocity autocorrelation function and ils power specirum and (b) the angular velocity autocorrelation function and the power specirum for differeni phases of CF4. (From Nose and Klein (6).)... 

Power spectra of the autocorrelation functions of the linear and angular velocities parallel and perpendicular to the C3 symmetrical axes have also been examined by Neusy et al. (32). In the rotator phase, there is good agreement with the Raman data (36). The calculated characteristic time (r4) for reorientation of the C3 axes from one [111] direction to another and also the reorientation time (r3) for rotation of molecules around the C3 axes were similar... 

The angular velocity and angular momentum acfs themselves are important to any dynamical theory of molecular liquids but are very difficult to extract directly from spectral data. The only reliable method available seems to be spin-rotation nuclear magnetic relaxation. (An approximate method is via Fourier transformation of far-infrared spectra.) The simulated torque-on acfs in this case become considerably more oscillatory, and, which is important, the envelope of its decay becomes longer-lived as the field strength increases. This is dealt with analytically in Section III. In this case, computer simulation is particularly useful because it may be used to complement the analytical theory in its search for the forest among the trees. Results such as these for autocorrelation functions therefore supplement our... 

The velocity and angular velocity autocorrelation functions become more oscillatoiy as the externally tqiplied torque increases in strength (Fig 3). Moving-frame component acfs such as (o (f)o (0)>/ have different time dependences, that is,... 

The nature of the excitation has a profound influence on the subsequent relaxation of molecular Uquid systems, as the molecular dynamics simulations show. This influence can be exerted at field-on equiUbrium and in decay transients (the deexdtation effect). GrigoUni has shown that the effect of high-intensity excitation is to slow the time decay of the envelope of such oscillatory functions as the angular velocity autocorrelation function. The effect of high-intensity pulses is the same as that of ultrafast (subpicosecond laser) pulses. The computer simulation by Abbot and Oxtoby shows that... 

When y - A2 the equivalent of the microscopic time y is = y/ -Decoupling effects are present when Uj = F. To obtain an approximate value of F we can use the experimental data as follows. First, we evaluate the value of decay of the oscillation envelopes of the angular velocity autocorrelation function as a function of Equation (14) shows that this is, approximately, a Lorentzian, the linewidth of which provides the approximate expression for F. The agreement with the numerical decoupling effect is quantitatively good when the ratio ai/uf is assumed to be equal to 8.5. Simple Markovian models cannot account for decoupling effects. 

Figure 6. (a) 2-Gilorobutane at 50 K, 6x6 site-site potential, angular velocity autocorrelation functions. Crosshatdiing indicates computer noise difference between R and S enantiomers. (—) Racemic mixture, (b) As for (a), under the influence of a strong field E, producing a torque — 63 XE in each molecule of the molecular dynamics sample. (1) (—) Racemic mixture (2) (—) R enantiomer. Ordinate Normalized correlation function abscissa time, ps. 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

Other orientational correlation coefficients can be calculated in the same way as the correlation coefficients that we have discussed already. Thus, the reorientational correlation coefficient of a single rigid molecule indicates the degree to which the orientation of a molecule at a time t is related to its orientation at time 0. The angular velocity autocorrelation function is the rotational equivalent of the velocity correlation function ... 

The rotational relaxation of water molecules is often discussed in terms of angular momentum autocorrelation functions (e g., Stillinger and Rahman 1972 Yoshii et al. 1998). For a flexible water model, a slightly different approach can also be used. In order to separate the various modes of molecular librations (hindered rotations) and intramolecular vibrations, the scheme proposed by Bopp (1986) and Spohr et al. (1988) can be employed. The instantaneous velocities of the two hydrogen atoms of every water molecule in the molecular center-of-mass system are projected onto the instantaneous unit vectors i) in the direction of the corresponding OH bond (ui and U2) ... 

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