Angular velocity correlation function equation

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 same result is obtained for linear molecules. Equation (26) suggests that a useful way to compare correlation functions with different L values is to plot[ilnC (t)J/L(L + 1) versus t. More importantly it shows that the angular velocity correlation function determines the lowest order contribution to CL(t). Before we look... 

According to Keyes and Kivelson 22 the collective orientational correlation function in the diffusion limit is given by equations (21) and (22). It has been argued 22 and to a certain extent verified experimentally 131, that the angular velocity correlation factor j2 is close to unity for ordinary liquids. 

Correlations by Computation of Molecular Dynamics. The power of modem computing systems has made it possible to solve the dynamical equations of motion of a model system of several hundred molecules, with fairly realistic interaction potentials, and hence by direct calculation obtain correlation functions for linear velocity, angular velodty, dipole orientation, etc. Rahman s classic paper on the motion of 864 atoms of model argon has stimulated a great amount of further work, of which we cite particularly that of Beme and Harpon nitrogen and carbon monoxide, and that of Rahman himself and Stillinger on water. ... 

In the MD method (Rahman, 1966 Verlet, 1967) one specifies the initial conditions, i.e., the positions and orientations of all molecules and their (angular) velocities, and one integrates the classical equations of motion numerically by means of some finite difference scheme. The choice of the time step is mainly determined by the error allowed and the time scale in which one is interested. Usually, one takes time steps of order 10 14 sec and follows the trajectory for some 103 to 105 steps. Along the whole or part of the trajectory all kinds of quantities can be averaged. The method is especially suitable for calculating time-dependent correlation functions that yield information on the dynamics of the system. An example is the so-called intermediate scattering function ... 

Distribution and Correlation Functions.—We consider a single spherical particle with position r and velocity v at time t in a concentrated dispersion of mean number density p. The distribution function measures the probability of finding a particle (the same or another particle) with position r" and velocity v" at time t". The osmotic equation of state is related to a time-averaged distribution function that depends on r alone, whereas the dynamic behaviour depends on time-dependent functions. A basic premise of statistical mechanics is that a time-average is equivalent to an ensemble average at fixed time the ensemble average is denoted by angular brackets (...). 

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