Self-propelled particles with selective attraction-repulsion interaction: from microscopic dynamics to coarse-grained theories

In this work we derive and analyse coarse-grained descriptions of self-propelled particles with selective attraction-repulsion interaction, where individuals may respond differently to their neighbours depending on their relative state of motion (approach versus movement away). Based on the formulation of a nonlinear Fokker-Planck equation, we derive a kinetic description of the system dynamics in terms of equations for the Fourier modes of the one-particle density function. This approach allows effective numerical investigation of the stability of solutions of the nonlinear Fokker-Planck equation. Further on, we also derive a hydrodynamic theory by performing a closure at the level of the second Fourier mode of the one-particle density function. We show that the general form of equations is in agreement with the theory formulated by Toner and Tu. The stability of spatially homogeneous solutions is analysed and the range of validity of the hydrodynamic equations is quantified. Finally, we compare our analytical predictions on the stability of the homogeneous solutions with results of individual-based simulations. They show good agreement for sufficiently large densities and non-negligible short-ranged repulsion. The results of the kinetic theory for weak short-ranged repulsion reveal the existence of a previously unknown phase of the model consisting of dense, nematically aligned filaments, which cannot be accounted for by the present hydrodynamics theory of the Toner and Tu type for polar active matter.