Relaxation time of a Brownian rotator in a potential with nonparabolic barriers

The extension of the Kramers theory of the escape rate of a Brownian particle from a potential well to the entire range of damping proposed by Mel’nikov and Meshkov [V.I. Mel’nikov, S.V. Meshkov, J. Chem. Phys. 85 (1986) 1018] is applied to the inertial rotational Brownian motion of a fixed axis rotator in a potential V(θ)=−K₁cos2θ−K₂cos4θ, where θ is the angle specifying the orientation of the rotator and K₁ and K₂ are constants. It is shown that in the neighbourhood of K₁∼4K₂ (flat barrier), the Mel’nikov-Meshkov method must suitably be adapted so that the effect of a nonparabolic barrier top can be correctly accounted for in the calculation of the relaxation time. The results obtained are compared with numerical calculations of the longest relaxation time (inverse smallest nonvanishing eigenvalue) using a matrix continued fraction algorithm and reasonable agreement is obtained for K₁≥4K₂ and all values of the dissipation parameter.