Brownian Motion in Periodic Potentials

Abstract

In this chapter we apply some of the methods discussed in Chap. 10 for solving the Kramers equation for the problem of Brownian motion in a periodic potential. As discussed below, this problem arises in several fields of science, for instance in physics, chemical physics and communication theory. Restricting ourselves to the one-dimensional case, we deal with particles which are kicked around by the Langevin forces and move in a one-dimensional periodic potential (Fig. 11.1). Because of the excitation due to the Langevin forces the particles may leave the well and go either to the neighboring left or right well or they may move in the course of time to other wells which are further away. For long enough times the particles will thus diffuse in both directions of the x axis. As shown in Sect. 11.7 this diffusion can be described by a diffusion constant D, if we wait long enough. Thus the mean-square displacement is given by

$$\langle\lbrack x(t)-x(0)\rbrack^2\rangle=2 D\ t$$

((11.1))

for large times t. (The particles are then distributed over many potential wells.)