High friction limit of the Kramers equation: The multiple time-scale approach

The purpose of the paper is to give a pedagogical introduction to the multiple time-scale technique, on the example of the high-friction limit of the Kramers equation. We begin with a discussion of the standard perturbation technique as presented in van Kampen's reference book, which will be shown to fail in the long-time Limit. Application of the multiple time-scale technique avoids these difficulties and leads to a uniform expansion in powers of the inverse of the friction. Analogy with the Chapman-Enskog expansion is discussed. (C) 1997 American Association of Physics Teachers.