Lattice model of active matter

Active systems -- collections of self-propelling particles with mutual interactions -- have been the subject of much research in recent years. This is because they often exhibit novel properties such as collective motion, or motility induced phase separation, which are absent in their passive counterparts.

Here, we present a lattice-based derivation of an N-particle Fokker-Plank equation for self-propelling particles with continuous orientation in arbitrary dimension. Our approach is equivalent to the Langevin formulation of active brownian motion. The effects of both exclusion and mutual alignment are considered. In the mean-field approximation, our approach leads to a deterministic Toner-Tu theory that incorporates exclusion. We analyze the linear stability of the disordered and ordered states, and compare our results with lattice-based simulations.