Landau kinetic equation for dry aligning active models

The Landau equation is a kinetic equation based on the weak coupling approximation of the interaction between the particles. In the framework of dry active matter this new kinetic equation relies on the weak coupling approximation of both the alignment strength and the magnitude of the angular noise, instead of the hypothesis of diluteness. Therefore, it is a kinetic equation bridging between the Boltzmann (Bertin et al 2006 Phys. Rev. E 74 022101), and the Smoluchowski (Baskaran et al 2010 J. Stat. Mech. P04019) approximations, and allowing analytical descriptions at moderate densities. The form of the equation presents non-linear and density dependent diffusions and advections fully derived by the microscopic equations of motions. Finally, implementing the BGL procedure (Peshkov et al 2014 Eur. Phys. J. Spec. Top. 223 1315-44), the parameters of the Toner-Tu equations are derived showing the appearance of linearly stable homogeneous ordered solutions and mimicking the results obtained from the Boltzmann approach.