Double bracket dissipation in kinetic theory for particles with anisotropic interactions

TL;DR

This paper develops a unified kinetic framework using double bracket dissipation to derive equations for anisotropic particle dynamics, including nonlocal Darcy's law, Lie-Darcy equations, and Smoluchowski models.

Contribution

It introduces a novel double bracket kinetic approach to derive various anisotropic particle dynamics from dissipative Vlasov equations.

Findings

Derived nonlocal Darcy's law for mass density.

Formulated Lie-Darcy continuum equations for density and orientation.

Connected kinetic models to Smoluchowski equations through moment closures.

Abstract

We derive equations of motion for the dynamics of anisotropic particles directly from the dissipative Vlasov kinetic equations, with the dissipation given by the double bracket approach (Double Bracket Vlasov, or DBV). The moments of the DBV equation lead to a nonlocal form of Darcy's law for the mass density. Next, kinetic equations for particles with anisotropic interaction are considered and also cast into the DBV form. The moment dynamics for these double bracket kinetic equations is expressed as Lie-Darcy continuum equations for densities of mass and orientation. We also show how to obtain a Smoluchowski model from a cold plasma-like moment closure of DBV. Thus, the double bracket kinetic framework serves as a unifying method for deriving different types of dynamics, from density--orientation to Smoluchowski equations. Extensions for more general physical systems are also discussed.