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 V K , 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 ChapmanEnskog expansion is discussed. Accepted for publication in American Journal of Physics February 2008