Autocorrelation function damping

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. 

C. The Rosch and Ratner Directly Damped Quantum Autocorrelation Function... 

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

It has been shown that the autocorrelation function (46) is the limit situation in the absence of damping of the autocorrelation function of the hydrogen bond within the indirect damping [8]. It must be emphasized that this autocorrelation function does not require for computation any particular caution because of its analytic character. Thus, it may be considered as a numerical reference for the computation involving representation I or III. 

Boulil et al. [8] have obtained the following expression for the quantum indirect damped autocorrelation function in representation II ... 

We may observe that for very weak damping, (5 a0, and the above autocorrelation function reduces to... 

Note that this last expression is nothing but the closed form [90] of the autocorrelation function obtained (as an infinite sum) in quantum representation III by Boulil et al.[87] in their initial quantum approach of indirect damping. Although the small approximation involved in the quantum representation III and avoided in the quantum representation II, both autocorrelation functions are of the same form and lead to the same spectral densities (as discussed later). 

The process X(t) is now fully specified since it is Gaussian and the first two moments are known. But it is not the same as the Wiener process determined by (3.1), because the autocorrelation function is more complicated than (IV.2.7a). In fact, X(t) is not even Markovian, owing to the fact that it is still described on the fine time scale belonging to the Rayleigh particle. On the coarse time scale only time differences much larger than the damping time l/y of the velocity are admitted,... 

With the above-mentioned simplifications, the equation of motion for the density time autocorrelation function reduces to the following form of a nonlinear equation of motion for a damped oscillator ... 

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