Autocorrelation function, momentum

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. 

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. 

In the canonical ensemble (P2) = 3kBTM and p M. In the microcanonical ensemble (P2) = 3kgT i = 3kBTMNm/(M + Nm) [49]. If the limit M —> oo is first taken in the calculation of the force autocorrelation function, then p = Nm and the projected and unprojected force correlations are the same in the thermodynamic limit. Since MD simulations are carried out at finite N, the study of the N (and M) dependence of (u(t) and the estimate of the friction coefficient from either the decay of the momentum or force correlation functions is of interest. Molecular dynamics simulations of the momentum and force autocorrelation functions as a function of N have been carried out [49, 50]. 

This result is interesting, since it gives the slip length as a function of parameters that can be measured experimentally or a priori, for simple systems in a linear approximation. The bulk shear viscosity can be approximated from the literature, and the monolayer density can be determined from optical techniques. To a first approximation, for rigidly adsorbed layers, the sliptime is related to the autocorrelation function of random momentum fluctuations in the film, given by [40]... 

The linear momentum or velocity autocorrelation function f(t), defined by... 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

We have already seen that even in the case of strong intermolecular interactions neither nor decay initially as exponentials. Gordon has been able to reproduce the decay of these latter functions in liquid CO and N2 by allowing for large angular displacements between interactions.58 However, Gordon s model incorrectly predicts the angular momentum autocorrelation function. 

Fig. 11. Angular momentum autocorrelation functions from the Stockmayer simulation of CO.

Fig. 13. Angular momentum autocorrelation functions from (a) the Stockmayer simulation of CO with a dipole moment of 1.172 Debye, and (b) the Lennard-Jones plus quadrupole-quadrupole simulation of N2.

In the preceding sections it was shown that the normalized linear and angular momentum autocorrelation functions, P(r) and Aj(t), are identical within experimental error with the corresponding directional autocorrelation functions,... 

As we saw in the previous sections, the normalized angular momentum autocorrelation function, Aj(t),... 

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. 

Fig. 31. Angular momentum autocorrelation functions from the modified Stock-mayer simulation of CO, the exponential memory, and the Gaussian memory I.

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. 

The simple expression (14,5) could lead to the erroneous conclusion that the momentum dependence of the autocorrelation function in tree approximation follows the Debye function, reproducing the result for a noninteracting chain. This conclusion is false, since the uncritical surface and thus both Nr and q2 nontrivially depend on q3. The effects show up in the region of large momenta q2 L... 

In the dilute region w < 1 the autocorrelation function directly crosses over from Debye-type behavior on scales q2Ji2 < 1 to the large momentum behavior as discussed above. For semidilute systems w intermediate regime, governed by the asymptotics of the Debye function,... 

Le Quere and Leforestier (1990, 1991) calculated the autocorrelation function directly using the same PES and found fair agreement in regard of the recurrence times while the amplitudes were in remarkable disagreement. This may be due to either deficiencies of the calculated PES or the neglect of nonzero total angular momentum states in the theory. If the recurrences in the autocorrelation function are rescaled, the quantum mechanically calculated spectrum agrees well with experiment. 

Let us then derive an expression for the matrix element of the flux operator in the coordinate representation, an expression we need in order to develop the time autocorrelation function of the flux operator in the coordinate representation. We use the axiom for the matrix element of the momentum operator in the coordinate representation, and obtain... 

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... 

In the dilute region tc < 1 the autocorrelation function directly crosses over from Debye-type behavior on scales < 1 to the large momentum... 

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