Activated rate processes: A relation between Hamiltonian and stochastic theories

Abstract

Kramers’ treatment of activated rate processes is based on the Langevin equation of motion for the escaping particle. The stochastic dynamics may be cast equivalently as the dynamics of a particle interacting bilinearly with a bath of harmonic oscillators. This paper explores the connection between the solutions of Kramers’ problem (and its generalization to include memory friction) obtained in the framework of these two approaches. We demonstrate their equivalence for the specific case of a parabolic barrier potential. The Hamiltonian representation is used to construct (a) a nontrivial eigenfunction of the Fokker–Planck equation which is generalized to include time dependent friction; (b) the Kramers’ stationary flux distribution function; (c) the stochastic separatrix.