Abstract

Collective dynamics can be observed among many animal species, and have given rise in the last decades to an active and interdisciplinary field of study. Such behaviours are usually modelled by active matter, in which each individual is self-driven and tends to align its velocity with that of its neighbours. In a classical model introduced by Vicsek & al., as well as in numerous related active matter models, a phase transition between chaotic behaviour at high temperature and global order at low temperature can be observed. Even though ample evidence of these phase transitions has been obtained for collective dynamics, from a mathematical standpoint, such active systems are not fully understood yet. Some progress has been achieved in the recent years under an assumption of mean-field interactions, however to this day, few mathematical results have been obtained for models involving purely local interactions. I will describe a lattice active particle system interacting locally to align their velocities, and give some elements on how to derive the hydrodynamic limit for such an out-of equilibrium system.