Velocity correlation functions, time behavior

Therefore, before describing the modification of the equilibrium FDT, we need to study in details the behavior of D(t). Note, however, that the integrated velocity correlation function [, Cvv(/) df takes on the meaning of a time-dependent diffusion coefficient only when the mean-square displacement increases without bounds (when the particle is localized, this quantity characterizes the relaxation of the mean square displacement Ax2 t) toward its finite limit Ax2(oo)). 

Here at = [ciqf/kT], a and qt are, respectively, the concentration and charge of species i (anions and cations). Vi 0)Vi(t)) and AFf(t)) are, respectively, the velocity correlation function (VCF) and mean-squared displacement in time of species i. The steady current behavior at long times in the step-on experiment (see above) means that ABf t)) becomes linearly dependent on time, giving the Nernst-Einstein equation that connects the low-/" conductivity cr (O) (=diffusion coefficients A for the translational motions of the ions (9,10) ... 

Boltzmann-Enskog approximation consists of recollision terms that describe correlated binary collisions. The effects of these processes on the behavior of time correlation functions have not yet been fully studied because the calculations involved are considerably more complicated. The problem where the effects of recollisions have been most extensively investigated is that of the velocity autocorrelation function, which is a simpler function than the dynamic structure factor. From this problem we can already see the kind of analysis involved in treating correlated collisions. ... 

Mobility in this region is dominated by short-time motion, typically < 2 ps. After that time, all correlation of molecular motion is lost due to frequent collisions with the cavity walls. The center-of-mass velocity autocorrelation function of the penetrant exhibits typical liquid-like behavior with a negative region due to velocity reversal when the penetrant hits the cavity wall [59]. This picture has recently been confirmed by Pant and Boyd [62] who monitored reversals in the penetrant s travelling direction when it hits the cavity walls. The details of the velocity autocorrelation function are not very sensitive to the force-field parameters used. On the other hand, the orientational correlation function of diatomic penetrants showed residuals of a gas-like behavior. Reorientation of the molecular axis does not have the signature of rotational diffusion, but rather shows some amount of free rotation with rotational correlation times of the order of a few tenths of a picosecond, although dependent in value on the Lennard-Jones radii of the penetrant s atoms. 

---