Chapman-Enskog derivation of the generalized Smoluchowski equation

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 effective generalized thermodynamical formalism.