Exact solution of Smoluchowski's equation for reorientational motion in Maier-Saupe potential

TL;DR

This paper derives an exact analytical solution to Smoluchowski's equation for reorientational motion in a symmetric Maier-Saupe potential, providing precise results across various barrier heights and highlighting the importance of transient dynamics.

Contribution

It presents the first exact solution of Smoluchowski's equation in a symmetric Maier-Saupe potential using confluent Heun functions, applicable to any barrier height.

Findings

Exact solution valid for all barrier heights

Agreement with existing methods at high barriers

Greater transition rates predicted at low barriers

Abstract

The analytic treatment of the non-inertial rotational diffusion equation, i.e., of the Smoluchowski's one (SE), in a symmetric genuinely double-well Maier-Saupe uniaxial potential of mean torque is considered. Such potential may find applications to reorientations of the fragments of structure in polymers and proteins. We obtain the exact solution of SE via the confluent Heun's function. The solution is uniformly valid for any barrier height. We apply the obtained solution to the calculation of the mean first passage time and the longitudinal correlation time and obtain their precise dependence on the barrier height. In the intermediate to high barrier (low temperature) region the results of our approach are in full agreement with those of the approach developed by Coffey, Kalmykov, D\'ejardin and their coauthors. In the low barrier (high temperature) region our results noticeably distinguish from the predictions of the literature formula and give appreciably greater values for the transition rates from the potential well. The reason is that the above mentioned formula is obtained in the stationary limit. We conclude that for very small barrier heights the transient dynamics plays a crucial role and has to be taken into account explicitly. When this requirement is satisfied (as, e.g, at the calculation of the longitudinal correlation time) we obtain absolute identity of our results with the literature formula in the whole range of barrier heights. The drawbacks of our approach are its applicability only to the symmetric potential and its inability to yield an analytical expression for the smallest non-vanishing eigenvalue.