Velocity correlation function calculation

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

We present ab-initio calculations of the velocity correlation function of the electronic states close to the Fermi energy, in a complex approximant and simple crystals. These calculations are the first numerical proof of the exis-... 

This expression reduces the calculations to the evaluation of the time-dependent velocity correlation function... 

The mobility of a macromolecule, constrained by other macromolecules, can be also calculated as (5.1). In the linear approximation, the zeroth normal co-ordinates of the macromolecule (equation (4.1), at z/jj = 0) define diffusive mobility of macromolecule. The one-sided Fourier transform velocity correlation function is determined by expression (4.15), so that we can write down the Fourier transform... 

Spectra of other dynamical variables like CoM velocities and zeolite window diameters can also be obtained by Fourier transformation of the appropriate time correlation functions. Calculation of spectra for different spatial components of a time dependent quantity provides useful information about the anisotropy of the corresponding motion. 

Figure 8. Instantaneous normal mode spectrum of liquid water. Solid, dashed, and dashed-dotted lines are calculated from the velocity correlation function, INM, and QNM, respectively. The system size in numerical simulations is 216.

Equation (239) may also be used to calculate the angular velocity correlation function (AVCF) in the fractional dynamics. From Eq. (239) with n = 1 and q= 0, we have... 

The complex rotational behavior of interacting molecules in the liquid state has been studied by a number of authors using MD methods. In particular we consider here the work of Lynden-Bell and co-workers [60-62] on the reorientational relaxation of tetrahedral molecules [60] and cylindrical top molecules [61]. In [60], both rotational and angular velocity correlation functions were computed for a system of 32 molecules of CX (i.e., tetrahedral objects resembling substituted methanes, like CBt4 or C(CH3)4) subjected to periodic boundary conditions and interacting via a simple Lennard-Jones potential, at different temperatures. They observe substantial departures of both Gj 2O) and Gj(() from predictions based on simple theoretical models, such as small-step diffusion or 7-diffusion [58]. Although we have not attempted to quantitatively reproduce their results with our mesoscopic models, we have found a close resemblance to our 2BK-SRLS calculations. Compare for instance our Fig. 13 with their Fig. 1 in [60]. 

It is of considerable interest to calculate quantum velocity correlation functions using CMD. In Fig. 10, the velocity (actually momentum) correlation function is shown for the nonlinear oscillator in Eq. (3.85) at /3 = 10. This simple test was again designed to compare the exact result with the CMD estimate. The latter is clearly very accurate. 

Figure 17. Plot of the centroid velocity correlation function for liquid neon. The solid line is the CMD result calculated with the centroid pseudopotential approximation, while the dashed line is the classical MD result. The self-diffusion constant is proportional to the time integral of the centroid velocity correlation functions.

Diffusion eoeffieients can be calculated directly from the velocity correlation functions or from mean square displacements, as ... 

In the molecular dynamics comparison of the full and the purely repulsive Lennard-Jones potentials, the velocity correlation function was calculated using both potentials in a variety of systems. However, the extremely close... 

Time correlation functions can be used in conjunction with the Green-Kubo relations to calculate the various transport coefficients in the system. For example, the self-diffusion coefficient D is related to the time integral of the velocity correlation function ... 

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

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. 

Fig. 1.14. Comparison of the MD calculations of the correlation functions of the translational velocity and angular momentum in liquid nitrogen [65]. The time is in units of 10-13 s.

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