Theory of activated rate processes: Exact solution of the Kramers problem

The Kramers theory for the escape rate of a Brownian particle from a potential well is extended to the full damping range. It is shown that the most adequate description of the underdamped Brownian motion in a deep potential well is provided by a Green function of the Fokker–Planck equation in the energy-position variables. The problem of lifetime of a particle in a single potential well is reduced to an integral equation in energy variable, with the Green function being the kernel of this equation. The straightforward solution by the Wiener–Hopf method yields an explicit expression for the lifetime, which describes the crossover from the extremely underdamped regime to that of a moderate damping. With the use of Kramer’s result for moderate-to-large damping an expression for the lifetime is presented, which holds at arbitrary damping. The problem of the rate of transitions between the two minima of a double-well potential is reduced to a system of two integral equations, which is also solved by the Wiener–Hopf method. An explicit expression for the relaxation time of nonequilibrium populations of the two minima is given.