Solutions of the Kramers Equation

Abstract

The Kramers equation is a special Fokker-Planck equation describing the Brownian motion in a potential. For a one-dimensional problem it is an equation for the distribution function in position and velocity space. This Kramers equation was derived and used by Kramers [1.17] to describe reaction kinetics. Later on it turned out that it had more general applicability, e.g., to such different fields as superionic conductors, Josephson tunneling junction, relaxation of dipoles, second-order phase-locked loops. These applications will be discussed in Chap. 11. For large damping constants the Kramers equation reduces to the Smoluchowski equation which is a special Fokker-Planck equation for the distribution function for the position coordinate only. In this chapter some of the well-known solutions for linear forces are presented. Next we shall derive a general solution of the Kramers equation in terms of matrix continued fractions for arbitrary forces. Expansion of these matrix continued-fraction solutions for large damping constants into powers of the inverse friction constant gives the Smoluchowski equation and its different correction terms. Whereas the position will become a slow variable and the velocity a fast variable in the high-friction limit, the energy will become a slow variable and the position (or velocity) a fast variable in the low-friction limit (see Sect. 8.3 for a discussion of slow and fast variables). In the low-friction limit the procedure depends on the topology of the energy surface in phase space, which in turn depends on the specific form of the potential.