Double-bracket dissipation in kinetic theory for particles with anisotropic interactions

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 non-local 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.