Chapman–Enskog derivation of the generalized Smoluchowski equation

✍ Scribed by Pierre-Henri Chavanis; Philippe Laurençot; Mohammed Lemou

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 318 KB
• Volume: 341
• Category: Article
• ISSN: 0378-4371
• DOI: 10.1016/j.physa.2004.04.102

✦ Synopsis

We use the Chapman-Enskog method to derive the Smoluchowski equation from the Kramers equation in a high friction limit. We consider two main extensions of this problem: we take into account a uniform rotation of the background medium and we consider a generalized class of Kramers equations associated with generalized free energy functionals. We mention applications of these results to systems with long-range interactions (self-gravitating systems, 2D vortices, bacterial populations, etc.). In that case, the Smoluchowski equation is non-local. In the limit of short-range interactions, it reduces to a generalized form of the Cahn-Hilliard equation. These equations are associated with an e ective generalized thermodynamical formalism.