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. With the exception of the linear force, special potentials are not treated in this chapter. Therefore, the low-friction limit is treated in Chap. 11 (Sects. 4, 6.3, 8.1, 9.1), where Brownian motion in a periodic potential is investigated.