Integral representation of exact solutions for the correlation times of rotators in periodic potentials — derivation of asymptotic expansions

The derivation of asymptotic expansions from the exact solution of the three term recurrence relations arising in the study of the Brownian movement in a periodic potential is discussed. The discussion is illustrated by showing how the exact formulae for the longitudinal and transverse correlation times of a single axis rotator with two equivalent sites, which have been previously given as a series of products of modified Bessel functions, may be rendered in integral form using Watson's integral formula for the product of two modified Bessel functions. The method of steepest descents is applied to these solutions in order to obtain rigourous asymptotic formulae for the correlation times in the high potential barrier limit. The analogous results for rotation in three dimensions in the Maier-Saupe potential are treated briefly.