Autocorrelation second-order orientational

Theoretical Models for Local Segmental Motions. The second order orientation autocorrelation function measured in this experiment is defined by ... 

Here 6(t) is the angle between the C-H bond vector at time zero and time t. This particular correlation function is referred to as a second order orientation autocorrelation function. 

This time is smaller than for a linear chain of Aymonomers but much larger than rd. Since D is very small, it is difficult to measure in a simulation and has not been done. The rotational diffusion time, which should be comparable to td, can be analyzed by studying the time autocorrelation function of the center-to-end vector R or the autocorrelation of the squares of the second-order spherical harmonics of the angles at which the principal axes of the ellipsoid are oriented with respect to a fixed-coordinate system. However even for small there was no linear region in the semilog plots of C t) for the times that can presently be simulated, making it impossible at present to test eq. (9.23). 

The values of these autocorrelation functions at times t = 0 and t = 00 are related to the two order parameters orientational averages of the second- and fourth-rank Legendre polynomial P2(cos/ ) and P4 (cos p). respectively, relative to the orientation p of the probe axis with respect to the normal to the local bilayer surface or with respect to the liquid crystal direction. The order parameters are defined as... 

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