Optical theorem for the Kramers’ problem

Abstract

Optical theorem formulation of the Kramers’ problem is proposed. A new technique to estimate the rate of escape in the Kramers model is developed, making use of analyticity relations analogous to those of the scattering amplitude appearing in particle physics. The formalism is shown to provide a very short derivation of the Mel'nikov–Meshkov solution for energy distribution functions in the barrier region, because of analyticity requirements alone. When the escape event is viewed as a causal one, appropriate analytic functions yielding the rate constant obey some relations in the Fourier space very analogous to those of the Kramers and Krönig for the dielectric function. While it seems to be rather a simplistic description, it reveals a linkage between the refractive index and the superionic conduction. We briefly discuss other relevant aspects as the ac conductivity of ionic solids and relationships between the activation energy of superionic conductors and dielectric constant quite similar with those reported by Wakamura (JPCS, 59 (1998) 591).