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

TL;DR

This paper introduces the multiple time-scale method to analyze the high friction limit of the Kramers equation, providing a pedagogical approach that overcomes limitations of standard perturbation techniques.

Contribution

It presents a detailed, accessible explanation of the multiple time-scale technique applied to the Kramers equation, highlighting its advantages over traditional perturbation methods.

Findings

Standard perturbation fails in long-time limit

Multiple time-scale method provides uniform expansion

Analogy with Chapman-Enskog expansion

Abstract

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 \refto{VK}, 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.