Correlation function, collision integral

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

According to (4.38), the correlation function (5.4) determines the collision integral of the relevant kinetic equation for the distribution of pairs. 

The parameter the diffusion collision integral, is a function of k T/e, where is the Boltzmann constant and e is a molecular energy parameter. Values of tabulated as a function of k T/e, have been published (Hirschfelder et al., 1964 Bird et al., 1960). Neufeld et al., (1972) correlated using a simple eight parameter equation that is suitable for computer calculations (see, also, Danner and Daubert, 1983 Reid et al., 1987). Values of a and e/k (which has units of kelvin) can be found in the literature—for only a few species—or estimated from critical properties (Reid et al., 1987 Danner and Daubert, 1983). The mixture a is calculated as the arithmetic average of the pure component values. The mixture e is taken to be the geometric average of the pure component values. 

When the integration time is long compared to the mean time T = A/v between successive collisions (which depends on the mean free path A and the mean relative velocity v) the total phase shift n(t) is the sum of many random phase shifts caused by collisions with different impact parameters (see Fig.3.9). The mean phase shift per collision depends on the correlation between the oscillation before and after the collisions, which can be described by the correlation function... 

Luminescence lifetime spectroscopy. In addition to the nanosecond lifetime measurements that are now rather routine, lifetime measurements on a femtosecond time scale are being attained with the intensity correlation method (124), which is an indirect technique for investigating the dynamics of excited states in the time frame of the laser pulse itself. The sample is excited with two laser pulse trains of equal amplitude and frequencies nl and n2 and the time-integrated luminescence at the difference frequency (nl - n2 ) is measured as a function of the relative pulse delay. Hochstrasser (125) has measured inertial motions of rotating molecules in condensed phases on time scales shorter than the collision time, allowing insight into relaxation processes following molecular collisions. 

The stochastic collision operator is assumed to be local in space and time, so that there are no correlations between the noise at different lattice sites or at different times. The delta functions in space and time have been replaced by b and /i , respectively, so that the double integral of (47) with respect to r and t over a small space-time region of size b h, matches the corresponding integral of (147). 

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