Non-Markovian diffusion over a parabolic potential barrier: influence of the friction-memory function

TL;DR

This paper analyzes how non-Markovian effects influence the probability of a particle crossing a parabolic barrier, using classical and quantum generalized Langevin equations, highlighting the role of the dominant root in the probability expression.

Contribution

It derives a general expression for over-passing probability incorporating memory effects via the dominant root of the characteristic function.

Findings

Asymptotic over-passing probability depends on a single dominant root.

Memory effects influence the probability through the characteristic equation.

Classical and quantum cases are both considered.

Abstract

The over-passing probability across an inverted parabolic potential barrier is investigated according to the classical and quantal generalized Langevin equations. It is shown that, in the classical case, the asymptotic value of the over-passing probability is determined by a single dominant root of the "characteristic function", and it is given by a simple expression. The expression for the over-passing probability is quite general, and details of dissipation mechanism and memory effects enter into the expression only through the dominant root of the characteristic equation.