Chapman-Enskog derivation of the generalized Smoluchowski equation

TL;DR

This paper derives a generalized Smoluchowski equation from the Kramers equation using Chapman-Enskog method, incorporating effects of background rotation and non-local interactions relevant to various physical and biological systems.

Contribution

It extends the derivation of the Smoluchowski equation to include background rotation and generalized free energy functionals, applicable to systems with long-range interactions.

Findings

Derived non-local Smoluchowski equation for long-range interactions

Reduced to a generalized Cahn-Hilliard equation for short-range interactions

Linked equations to an effective thermodynamical formalism

Abstract

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,...). 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.